Indefinitely that is to know as much as they can without propounding to themselves any limited question or they enquire into the Cause of some determined Appearance or endeavour to find out the certainty of something in question as what is the cause of Light of Heat of Gravity of a Figure propounded and the like or in what Subiect any propounded Accident is inhaerent or what may conduce most to the generation of some propounded Effect from many Accidents or in what manner particular Causes ought to be compounded for the production of some certaine Effect Now according to this variety of things in question sometimes the Analyticall Method is to be used and sometimes the Syntheticall 4 But to those that search after Science indefinitely which consists in the knowledge of the Causes of all things as far forth as it may be attained and the Causes of Singular things are compounded of the Causes of Universall or Simple things it is necessary that they know the Causes of Universall things or of such Accidents as are common to all Bodies that is to all Matter before they can know the Causes of Singular things that is of those Accidents by which one thing is distinguished from another And againe they must know what those Universall things are before they can know their Causes Moreover seeing Universall things are contained in the Nature of Singular things the knowledge of them is to be acquired by Reason that is by Resolution For example if there be propounded a Conception or Idea of some Singular thing as of a Square this Square is to be resolved into a Plain terminated with a certaine number of equall and straight lines and right angles For by this Resolution we have these things Universall or agreeable to all Matter namely Line Plain which containes Superficies Terminated Angle Straightness Rectitude and Equality and if we can find out the Causes of these we may compound them all together into the Cause of a Square Againe if any man propound to himselfe the Conception of Gold he may by Resolving come to the Ideas of Solid Visible Heavy that is tending to the Center of the Earth or downwards and many other more Universall then Gold it selfe and these he may Resolve againe till he come to such things as are most Universall And in this manner by Resolving continually we may come to know what those things are whose Causes being first known severally and afterwards compounded bring us to the Knowledge of Singular things I conclude therefore that the Method of attaining to the Universall Knowledge of Things is purely Analyticall 5 But the Causes of Universall things of those at least that have any Cause are manifest of themselues or as they say commonly known to Nature so that they need no Method at all for they have all but one Universall Cause which is Motion For the variety of all Figures arises out of the variety of those Motions by which they are made and Motion cannot be understood to have any other Cause besides Motion nor has the Variety of those things we perceive by Sense as of Colours Sounds Savours c. any other Cause then Motion residing partly in the Objects that work upon our Senses and partly in our selves in such manner as that it is manifestly some kind of Motion though we cannot without Ratiocination come to know what kind For though many cannot understand till it be in some sort demonstrated to them that all Mutation consists in Motion yet this happens not from any obscurity in the thing it selfe for it is not intelligible that any thing can depart either from Rest or from the Motion it has except by Motion but either by having their Naturall Discourse corrupted with former Opinions received from their Masters or else for this that they do not at all bend their mind to the enquiring out of Truth 6 By the Knowledge therefore of Universalls and of their Causes which are the first Principles by which we know the 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 of things we have in the first place their Definitions which are nothing but the explication of our Simple Conceptions For example he that has a true Conception of Place cannot be ignorant of this Definition Place is that space which is possessed or filled adaequately by some Body and so he that conceives Motion aright cannot but know that Motion is the privation of one Place and the acquisition of another In the next place we have their Generations or Descriptions as for example that a Line is made by the Motion of a Point Superficies by the Motion of a Line and one Motion by another Motion c. It remains that we enquire what Motion begets such and such Effects as what Motion makes a Straight line and what a Circular what Motion thrusts what drawes and by what way what makes a thing which is seen or heard to be seen or heard sometimes in one manner sometimes in another Now the Method of this kind of Enquiry is Compositive For first we are to observe what Effect a Body moved produceth when we consider nothing in it besides its Motion and we see presently that this makes a Line or length next what the Motion of a long Body produces which we find to be Superficies and so forwards till we see what the Effects of Simple Motion are and then in like manner we are to observe what proceeds from the Addition Multiplication Substraction and Division of these Motions and what Effects what Figures and what Properties they produce from which kind of Contemplation sprung that part of Philosophy which is called Geometry From this consideration of what is produced by Simple Motion we are to passe to the consideration of what Effects one Body moved worketh upon another and because there may be Motion in all the severall parts of a Body yet so as that the whole Body remain still in the same place we must enquire first what Motion causeth such and such Motion in the whole that is when one Body invades another Body which is either at Rest or in Motion what way and with what swiftnesse the invaded Body shall move and again what Motion this second Body will generate in a third and so forwards From which Contemplation shall be drawn that part of Philosophy which treats of Motion In the Third place we must proceed to the Enquiry of such Effects as are made by the Motion of the Parts of any Body as how it comes to passe that things when they are the same yet seeme not to be the same but changed And here the things we search after are sensible Qualities such as Light Colour Transparency Opacity Sound Odour Savour Heat Cold and the like which because they cannot be known till we know the Causes of Sense it selfe therefore the consideration of the Causes of Seeing Hearing Smelling Tasting and Touching belongs to this third place and all those qualities and Changes above mentioned are to be referred to
which make equal Angles on either side of the Perpendicular be produced to the Tangent they will be equal 12 There is in Euclide a Definition of Straight-lined Parallels but I do not find that Parallels in general are any where defined and therefore for an Universal definition of them I say that Any two lines whatsoever Straight or Crooked as also any two Superficies are PARALLEL when two equal straight lines wheresoever they fall upon them make always equal Angles with each of them From which Definition it follows First that any two straight lines not enclined opposite wayes falling upon two other straight lines which are Parallel and intercepting equal parts in both of them are themselves also equal and Parallel As if AB and CD in the third Figure enclined both the same way fall upon the Parallels AC and BD and AC and BD be equal AB and CD will also be equal and Parallel For the Perpendiculars BE and DF being drawn the Right Angles EBD and FDH will be equal Wherefore seeing EF and BD are parallel the angles EBA and FDC will be equal Now if DC be not equal to BA let any other straight line equal to BA be drawn from the point D which seeing it cannot fall upon the point C let it fall upon G. Whereore AG will be either greater or less then BD and therefore the angles EBA and FDG are not equal as was supposed Wherefore AB and CD are equal which is the first Again because they make equal Angles with the Perpendiculars BE and DF therefore the Angle CDH will be equal to the Angle ABD and by the Definition of Parallels AB and CD will be parallel which is the second That Plain which is included both wayes within parallel lines is called a PARALLELOGRAM 1 Corollary From this last it follows That the Angles ABD and CDH are equal that is that a straight line as BH falling upon two Parallels as AB and CD makes the internal Angle ABD equal to the external and opposite Angle CDH 2 Coroll And from hence again it follows that a straight line falling upon two Parallels makes the alternate Angles equal that is the Angle AGF in the fourth figure equal to the Angle GFD For seeing GFD is equal to the external opposite Angle EGB it will be also equal to its vertical Angle AGF which is alternate to GFD 3 Coroll That the internal Angles on the same side of the line FG are equal to two Right Angles For the Angles at F namely GFC and GFD are equal to two Right Angles But GFD is equal to its alternate Angle AGF Wherefore both the Angles GFC and AGF which are internal on the same side of the line FG are equal to two Right Angles 4 Coroll That the three Angles of a straight-lined plain Triangle are equal to two Right Angles and any side being produced the external Angle will be equal to the two opposite internal Angles For if there be drawn by the Vertex of the plain Triangle ABC figure 5. a Parallel to any of the sides as to AB the Angles A and B will be equal to their alternate Angles E F the Angle C is common But by the 10th Article the three Angles E C and F are equal to two Right Angles and therefore the three Angles of the Triangle are equal to the same which is the first Again the two Angles B and D are equal to two Right Angles by the 10th Article Wherefore taking away B there will remain the Angles A and C equal to the Angle D which is the second 5 Coroll If the Angles A and B be equal the sides AC and CB will also be equal because AB and EF are parallel And on the contrary if the sides AC and CB be equal the Angles A and B will also be equal For if they be not equal let the Angles B and G be equal Wherefore seeing GB and EF are parallels and the Angles G and B equal the sides GC and CB will also be equal and because CB and AC are equal by supposition CG and CA will also be equal which cannot be by the 11th Article 6 Coroll From hence it is manifest that if two Radii of a Circle be connected by a straight Line the Angles they make with that connecting Line will be equal to one another and if there be added that segment of the Circle which is subtended by the same line which connects the Radii then the Angles which those Radii make with the circumference wil also be equal to one another For a straight line which subtends any Arch makes equal Angles with the same because if the Arch and the Subtense be divided in the middle the two halves of the segment wil be congruous to one another by reason of the Uniformity both of the Circumference of the Circle and of the straight Line 13 Perimeters of Circles are to one another as their Semidiameters are For let there be any two Circles as in the first figure BCD the greater and EFG the lesser having their common Center at A and let their Semidiameters be AC and AE I say AC has the same proportion to AE which the Perimeter BCD has to the Perimeter EFG For the magnitude of the Semidiameters AC and AE is determined by the distances of the points C and E from the Center A and the same distances are acquired by the uniform motion of a point from A to C in such manner that in equal times the distances acquired be equal But the Perimeters BCD and EFG are also determined by the same distances of the points C and E from the Center A and therefore the Perimeters BCD and EFG as well as the Semidiameters AC and AE have their magnitudes determined by the same cause which cause makes in equal times equal spaces Wherefore by the 13 Chapter and 6th Article the Perimeters of Circles and their Semidiameters are Proportionals which was to be proved 14 If two straight Lines w ch cōstitute an Angle be cut by straight-lined Parallels the intercepted Parallels will be to one another as the parts w ch they cut off frō the Vertex Let the straight lines AB and AC in the 6 figure make an Angle at A be cut by the two straight-lined Parallels BC and DE so that the parts cut off from the Vertex in either of those Lines as in AB may be AB and AD. I say the Parallels BC and DE are to one another as the parts AB and AD. For let AB be divided into any number of equal parts as into AF FD DB and by the points F and D let FG and DE be drawn Parallel to the base BC and cut AC in G and E and again by the points G and E let other straight lines be drawn Parallel to AB and cut BC in H and I. If now the point A be understood to be moved uniformly over AB and in the
same time B be moved to C and all the points F D and B be moved uniformly and with equal Swiftness over FG DE and BC then shall B pass over BH equal to FG in the same time that A passes over AF and AF and FG will be to one another as their Velocities are and when A is in F D will be in K when A is in D D will be in E and in what manner the point A passes by the points F D and B in the same manner the point B will pass by the points H I and C the straight lines FG DK KE BH HI IC are equal by reason of their Parallelisme and therefore as the velocitie in AB is to the velocity 〈◊〉 BC so is AD to DE but as the velocity in AB is to the velocity in BC so is AB to BC that is to say all the Parallels will be severally to all the parts cut off from the Vertex as AF is to FG. Wherefore AF. GF AD. DE AB BC are Proportionals The Subtenses of equal Angles in different Circles as the straight lines BC and FE in the 1 figure are to one another as the Arches which they subtend For by the 8th Article the Arches of equal Angles are to one another as their Perimeters are and by the 13th Art the Perimeters as their Semidiameters But the the Subtenses BC and FE are parallel to one another by reason of the equality of the Angles which they make with the Semidiameters and therefore the same Subtenses by the last precedent Article will be proportional to the Semidiameters that is to the Perimeters that is to the Arches which they subtend 15 If in a Circle any number of equal Subtenses be placed immediatly after one another and straight lines be drawn from the extreme point of the first Subtense to the extreme points of all the rest The first Subtense being produced will make with the second Subtense an external Angle double to that which is made by the same first Subtense and a Tangent to the Circle touching it in the extreme point thereof and if a straight line which subtends two of those Arches be produced it will make an external Angle with the third Subtense triple to the Angle which is made by the Tangent with the first Subtense and so continually For with the Radius AB in the 7th figure let a circle be described in it let any number of equal Subtenses BC CD DE be placed also let BD BE be drawn by producing BC BD BE to any distance in G H and I let them make Angles with the Subtenses which succeed one another namely the external Angles GCD and HDE Lastly let the Tangent KB be drawn making with the first Subtense the Angle KBC I say the Angle GCD is double to the Angle KBC and the Angle HDE triple to the same Angle KBC For if AC be drawn cutting BD in M and from the point C there be drawn LC perpendicular to the same AC then CL and MD will be parallel by reason of the right Angles at C and M and therefore the alterne Angles LCD and BDC wil be equal as also the Angles BDC and CBD will be equal because of the equality of the straight lines BC and CD Wherefore the Angle GCD is double to either of the Aâ—gles CBD or CDB and therefore also the Angle GCD is double to the Angle LCD that is to the Angle KBC Again CD is parallel to BE by reason of the equality of the Angles CBE and DEB and of the straight lines CB and DE and therefore the Angles GCD and GBE are equal and consequently GBE as also DEB is double to the Angle KBC But the external Angle HDE is equal to the two internal DEB and DBE and therefore the Angle HDE is triple to the Angle KBC c. which was to be proved 1 Corollary From hence it is manifest that the Angles KBC and CBD as also that all the Angles that are comprehended by two straight lines meeting in the circumference of a Circle and insisting upon equal Arches are equal to one another 2 Coroll If the Tangent BK be moved in the Circumference with Uniform motion about the Center B it will in equal times cut off equal Arches and will pass over the whole Perimeter in the same time in which it self describes a semiperimeter about the Center B. 3 Coroll From hence also we may understand what it is that determines the bending or Curvation of a straight line into the circumference of a Circle namely that it is Fraction continually encreasing in the same manner as NuÌ„bers from One upwards encrease by the continual addition of Unity For the indefinite straight Line KB being broken in B according to any Angle as that of KBC again in C according to a double Angle and in D according to an Angle which is triple and in E according to an Angle which is quadruple â—o the first Angle and so continually there will be described a figure which will indeed be rectilineal if the broken parts be considered as haâ—ing magnitude but if they be understood to be the least tâ—aâ— can 〈◊〉 â—â—at is as so many Points then the figure described will â—ot be rectilineal but a Circle whose Circumference wâ—… broken line 4 〈…〉 been said in this present Article it may 〈…〉 Angle in the center is double to an Angle in the Circumference of the same Circle if the intercepted Arches be equal For seeing that straight Line by whose motion an Angle is determined passes over equal Arches in equal times as well from the Center as from the Circumference and while that which is from the Circumference is passing over half its own Perimeter it passes in the same time over the whole Perimeter of that which is from the Center the Arches w ch it cuts off in the Perimeter whose Center is A wil be double to those which it makes in its own Semiperimeter whose Center is B. But in equal Circles as Arches are to one another so also are Angles It may also be demonstrated that the external Angle made by a Subtense produced and the next equal Subtense is equal to an Angle from the Center insisting upon the same Arch As in the last Diagram the Angle GCD is equal to the Angle CAD For the external Angle GCD is double to the Angle CBD and the Angle CAD insisting upon the same Arch CD is also double to the same Angle CBD or KBC 16 An Angle of Contingence if it be compared with an Angle simply so called how little soever has such proportion to it as a Point has to a Line that is no proportion at all nor any quantity For first an Angle of coÌ„tingence is made by coÌ„tinual flexion so that in the generation of it there is no circular motion at all in which consists the nature of an Angle simply so called and therefore it cannot be
uniformly the Straight line P C. Seeing therefore the two Straight lines A P and P C are described in the time A E with the same encrease of Impetus wherewith the Crooked line A B C is described in the same time A E that is seeing the Line A P C and the Line A B C are transmitted by the same Body in the same Time with equal Velocities the Lines themselves are equal which was to be demonstrated By the same method if any of the Semiparabolasters in the Table of the 3d Article of the precedent Chapter be exhibited may be found a Straight line equal to the Crooked line thereof namely by dividing the Diameter into two equal parts and proceeding as before Yet no man hitherto hath compared any Crooked with any Straight Line though many Geometricians of every Age have endeavoured it But the cause why they have not done it may be this that there being in Euclide no Definition of Equality nor any mark by which to judge of it besides Congruity which is the 8th Axiome of the first Book of his Elements a thing of no use at all in the comparing of Straight and Crooked and others after Euclide except Archimedes and Apollonius and in our time Boâ—aâ—entura thinking the industry of the Ancients had reached to all that was to be done in Geometry thought also that all that could be propounded was either to be deduced from what they had written or else that it was not at all to be done It was therefore disputed by some of those Ancients themselves whether there might be any Equality at all between Crooked and Straight Lines Which question Archimedes who assumed that some Straight lineâ— was equal to the Circumference of a Circle seems to have despised as he had reason And there is a late Writer that granteth that between a Straight and a Crooked Line there is Equality but now now sayes he since the fall of Adam without the special assistance of Divine Grace it is not to be found CHAP. XIX Of Angles of Incidence and Reflection equal by supposition 1 If two straight lines falling upon another straight line be parallel the lines reflected from them shall also be parallel 2 If two straight lines drawn from one point fall upon another straight line the lines reflected from them if they be drawn out the other way will meâ—t in an angle equal to the angle made by the lines of Incidence 3 If two straight parallel lines drawn not oppositely but from the same parts fall upon the Circumference of a Circle the lines reflected from them if produced they meet within the Circle will make an angle double to that which is made by two straight lines drawn from the Center to the points of Incidence 4 If two straight lines drawn from the same point without a Circle fall upon the Circumference and the lines reflected from them being produced meet within the Circle they will make an angle equal to twice that angle which is made by two straight lines drawn from the Center to the points of Incidence together with the angle which the incident lines themselves make 5 If two straight lines drawn from one point fall upon the concave Circumference of a Circle and the angle they make be less then twice the angle at the Center the lines reflected from them and meeting within the Circle will make an angle which being added to the angle of the incident lines will be equal to twice the angle at the Center 6 If through any one point two unequal Chords be drawn cutting one another and the Center of the Circle be not placed between them and the lines reflected from them concurre wheresoever there cannot through the point through which the two former lines were drawn be drawn any other straight line whose reflected line shall pass through the common point of the two former lines reflected 7 In equal Chords the same is not true 8 Two points being given in the Circumference of a Circle to draw two straight lines to them so as that their reflected lines may contain any angle given 9 If a straight line falling upon the Circumference of a Circle be produced till it reach the Semidiameter and that part of it which is intercepted between the Circumference and the Semidiameter be equal to that part of the Semidiameter which is between the point of concourse the center the reflected line will be parallel to the Semidiameter 10 If from a point within a Circle two straight lines be drawn to the Circumference and their reflected lines meet in the Circumference of the same Circle the angle made by the reflected lines will be a third part of the angle made by the incident lines WHether a Body falling upon the superficies of another Body and being reflected from it do make equal angles at that superficies it belongs not to this place to dispute being a knowledge which depends upon the natural causes of Reflection of which hitherto nothing has been said but shall be spoken of hereafter In this place therefore let it be supposed that the angle of Incidence is equal to the angle of Reflection that our present search may be applyed not to the finding out of the causes but some consequences of the same I call an Angle of Incidence that which is made between a straight line and another line straight or crooked upon which it falls and which I call the Line Reflecting and an Angle of Reflection equal to it that which is made at the same point between the straight line which is reflected and the line reflecting 1 If two straight lines which fall upon another straight line be be parallel their reflected lines shall be also parallel Let the two straight lines AB and CD in the 1 figure which fall upon the straight line EF at the points B and D be parallel and let the lines reflected from them be BG and DH I say BG and DH are also parallel For the angles ABE and CDE are equal by reason of the parallellelisme of AB and CD and the angles GBF and HDF are equal to them by supposition for the lines BG and DH are reflected from the lines AB and CD Wherefore BG and DH are parallel 2 If two straight lines drawn from the same point fall upon another straight line the lines reflected from them if they be drawn out the other way will meet in an angle equal to the angle of the Incident lines From the point AC in the 2d figure let the two straight lines AB and AD be drawn and let them fall upon the straight line EK at the points B and D and let the lines BI and DG be reflected from them I say IB and GD do converge and that if they be produced on the other side of the line EK they shall meet as in F and that the angle BFD shal be equal to the angle BAD For the angle of Reflection IBK is equal to the angle
of Incidence ABE and to the angle IBK its vertical angle EBF is equal and therefore the angle ABE is equal to the angle EBF Again the angle ADE is equal to the angle of Reflection GDK that is to its vertical angle EDF and therefore the two angles ABD and ADB of the triangle ABD are one by one equal to the two angles FBD and FDB of the triangle FBD Wherfore also the third angle BAD is equal to the third angle BFD which was to be proved Corollary 1. If the straight line AF be drawn it will be perpendicular to the straight line EK For both the angles at E will be equal by reason of the equality of the two angles ABE and FBE and of the two sides AB and FB Corollary 2. If upon any point between B and D there fall a straight line as AC whose reflected line is CH this also produced beyond C will fall upon F which is evident by the demonstration above 3 If from two points taken without a Circle two straight parallel lines drawn not oppositely but from the same parts fall upon the Circumference the lines reflected from them if produced they meet within the Circle will make an angle double to that which is made by two straight lines drawn from the Center to the points of Incidence Let the two straight parallels AB and DC in the 3d figure fall upon the Circumference BC at the points B and C and let the Center of the Circle be E and let AB reflected be BF and DC reflected be CG and let the lines FB and GC produced meet within the Circle in H and let EB and EC be connected I say the angle FHG is double to the angle BEC For seeing AB and DC are parallels and EB cuts AB in B the same EB produced will cut DC somewhere let it cut it in D let DC be produced howsoever to I and let the intersection of DC BF be at K. The angle therefore ICH being external to the triangle CKH will be equal to the two opposite angles CKH and CHK Again ICE being external to the triangle CDE is equal to the two angles at D and E. Wherefore the angle ICH being double to the angle ICE is equal to the angles at D and E twice taken and therefore the two angles CKH and CHK are equal to the two angles at D and E twice taken But the angle CKH is equal to the angles D and ABD that is D twice taken for AB and DC being parallels the altern angles D and ABD are equal Wherefore CHK that is the angle FHG is also equal to the angle at E twice taken which was to be proved Corollary If from two points taken within a circle two straight parallels fall upon the circumference the lines reflected from them shall meet in an angle double to that which is made by two straight lines drawn from the center to the points of Incidence For the parallels LB and IC falling upon the points B and C are reflected in the lines BH and CH and make the angle at H double to the angle at E as was but now demonstrated 4 If two straight lines drawn from the same point without a circle fall upon the circumference and the lines reflected from them being produced meet within the circle they will make an angle equal to twice that angle which is made by two straight lines drawn from the center to the points of Incidence together with the angle which the incident lines themselves make Let the two straight lines AB and AC in the 4th figure be drawn from the point A to the circumference of the circle whose center is D and let the lines reflected from them be BE and CG and being produced make within the circle the angle H also let the two straight lines DB and DC be drawn from the center D to the points of Incidence B and C. I say the angle H is equal to twice the angle at D together with the angle at A. For let AC be produced howsoever to I. Therefore the angle CH which is external to the triangle CKH will be equal to the two angles GKH and CHK Again the angle ICD which is external to the triangle CLD wil be equal to the two angles CLD and CDL But the angle ICH is double to the angle ICD and is therefore equal to the angles CLD and CDL twice taken Wherefore the angles CKH and CHK are equal to the angles CLD and CDL twice taken But the angle CLD being external to the triangle ALB is equal to the two angles LAB LBA consequently CLD twice taken is equal to LAB LAB twice taken Wherefore CKH CHK are equal to the angle CDL together with LAB and LBA twice taken Also the angle CKH is equal to the angle LAB once and ABK that is LBA twice taken Wherefore the Angle CHK is equal to the remaining angle CDL that is to the angle at D twice taken and the angle LAB that is the angle at A once taken which was to be proved Corollary If two straight converging lines as IC and MB fall upon the concave circumference of a circle their reflected lines as CH and BH will meet in the angle H equal to twice the angle D together with the angle at A made by the â—ncident lines produced Or if the Incident lines be HB and IC whose reflected lines CH and BM meet in the point N the angle CNB will be equal to twice the angle D together with the angle CKH made by the lines of Incidence For the angle CNB is equal to the angle H that is to twice the angle D together with the two angles A and NBH that is KBA But the angles KBA and A are equal to the angle CKH Wherefore the angle CNB is equal to twice the angle D together with the angle CKH made by the lines of Incidence IC and HB produced to K. 5 If two straight lines drawn from one point fall upon the concave circumference of a circle and the angle they make be lesse then twice the angle at the center the lines reflected from them and meeting within the circle will make an angle which being added to the angle of the incident lines will be equal to twice the angle at the center Let the two Lines AB and AC in the 5th figure drawn from the point A fall upon the concave circumference of the circle whose center is D let their reflected Lines BE and CE meet in the point E also let the angle A be less then twice the angle D. I say the angles A and E together taken are equal to twice the angle D. For let the straight Lines AB and EC cut the straight Lines DC and DB in the points G and H and the angle BHC will be equal to the two angles EBH and E also the same angle BHC will be equal to the two angles D and DCH
and in like manner the angle BGC will be equal to the two angles ACD A the same angle BGC will be also equal to the two angles DBG and D. Wherefore the four angles EBH E ACD and A are equal to the four angles D DCH DBG and D. If therefore equals be taken away on both sides namely on one side ACD and EBH and on the other side DCH and DBG for the angle EBH is equal to the angle DBG and the angle ACD equal to the angle DCH the remainders on both sides will be equal namely on one side the angles A and E and on the other the angle D twice taken Wherefore the angles A and E are equal to twice the angle D. Corollary If the angle A be greater then twice the angle D their reflected â—â—ines will diverge For by the Corollary of the third Proposition if the angle A be equal to twice the angle D the reflected Lines BE and CE will be parallel and if it be lesse they will concurre as has now been demonstrated and therefore if it be greater the reflected Lines BE and CE will diverge and consequently if they be produced the other way they will concurre and make an angle equal to the excesse of the angle A above twice the angle D as is evident by the fourth Article 6 If through any one point two unequal chords be drawn cutting one another either within the circle or if they be produced without it and the center of the circle be not placed between them and the Lines reflected from them concurre wheresoever there cannot through the point through which the former Lines were drawn be drawn another straight Line whose reflected Line shall passe through the point where the two former reflected Lines concurre Let any two unequal chords as BK and CH in the 6th Figure be drawn through the point A in the circle BC and let their reflected Lines BD and CE meet in F and let the center not be between AB and AC and from the point A let any other straight Line as AG be drawn to the circumference between B and C. I say GN which passes through the point F where the reflected Lines BD and CE meet will not be the reflected Line of AG. For let the arch BL be taken equal to the arch BG and the straight Line BM equal to the straight Line BA and LM being drawn let it be produced to the circuÌ„mference in O. Seeing therefore BA and BM are equal and the arch BL equal to the arch BG and the angle MBL equal to the angle ABG AG and ML will also be equal and producing GA to the circumference in I the whole lines LO and GI will in like manner be equal But LO is greater then GFN as shall presently be demonstrated and therefore also GI is greater then GN Wherefore the angles NGC and IGB are not equal Wherefore the Line GFN is not reflected from the Line of Incidence AG and consequently no other straight Line besides AB and AC which is drawn through the point A and faâ—ls upon the circumference BC can be reflected to the point F which was to be demonstrated It remains that I prove LO to be greater then GN which I shall do in this manner LO and GN cut one another in P and PL is greater then PG. Seeing now LP PG PN PO are proportionals therefore the two Extremes LP and PO together taken that is LO are greater then PG and PN together taken that is GN which remained to be proved 7 But if two equal chords be drawn through one point within a circle and the Lines reflected from them meet in another point then another straight Line may be drawn between them through the former point whose reflected Line shall pass through the later point Let the two equal chords BC and ED in the 7th figure cut one another in the point A within the circle BCD and let their reflected Lines CH and DI meet in the point F. Then dividing the arch CD equally in G let the two chords GK and GL be drawn through the points A and F. I say GL will be the Line reflected from the chord KG For the four chords BC CH ED and DI are by supposition all equal to one another and therefore the arch BCH is equal to the arch EDI as also the angle BCH to the angle EDI the angle AMC to its vertical angle FMD and the straight Line DM to the straight Line CM and in like manner the straight Line AC to the straight Line FD and the chords CG and GD being drawn will also be equal as also the angles FDG and ACG in the equal Segments GDI and GCB Wherefore the straight Lines FG and AG are equal and therefore the angle FGD is equal to the angle AGC that is the angle of Incidence equal to the angle of Reflection Wherefore the line GL is reflected from the incident Line KG which was to be proved Corollary By the very sight of the figure it is manifest that if G be not the middle point between C and D the reflected Line GL will not pass through the point F. 8 Two points in the circumference of a circle being given to draw two straight Lines to them so as that their reflected Lines may be parallel or contain any angle given In the circumference of the circle whose center is A in the 8th figure let the two points B and C be given and let it be required to draw to them from two points taken without the circle two incident Lines so that their reflected Lines may first be parallel Let AB and AC be drawn as also any incident Line DC with its reflected Line CF and let the angle ECD be made double to the angle A and let HB be drawn parallel to EC and produced till it meet with DC produced in I. Lastly producing AB indefinitely to K let GB be drawn so that the angle GBK may be equal to the angle HBK and then GB will be the reflected Line of the incident Line HB I say DC and HB are two incident Lines whose reflected Lines CF and BG are parallel For seeing the angle ECD is double to the angle BAC the angle HIC is also by reason of the parallels EC and HI double to the same BAC Therefore also FC and GB namely the lines reflected from the incident lines DC and HB are parallel Wherefore the first thing required is done Secondly let it be required to draw to the points B C two straight lines of Incidence so that the lines reflected from them may contain the given angle Z. To the angle ECD made at the point C let there be added on one side the angle DCL equal to half Z and on the other side the angle ECM equal to the angle DCL and let the straight Line BN be drawn parallel to the straight line CM and let the angle KBO be made equal to the
angle NBK which being done BO will be the Line of Reflection from the Line of Incidence NB. Lastly from the incident Line LC let the reflected Line CO be drawn cutting BO at O and making the angle COB I say the angle COB is equal to the angle Z. Let NB be produced till it meet with the straight line LC produced in P. Seeing therefore the angle LCM is by construction equal to twice the angle BAC together with the angle Z the angle NPL which is equal to LCM by reason of the parallels NP and MC will also be equal to twice the same angle BAC together with the angle Z. And seeing the two straight lines OC and OB fall from the point O upon the points C and B and their reflected lines LC and NB meet in the point P the angle NPL will be equal to twice the angle BAC together with the angle COP But I have already proved the angle NPL to be equal to twice the angle BAC together with the angle Z. Therefore the angle COP is equal to the angle Z Wherefore Two points in the circumference of a Circle being given I have drawn c. which was to be done But if it be required to draw the incident Lines from a point within the circle so that the Lines reflected from them may contain an angle equal to the angle Z the same method is to be used saving that in this case the angle Z is not to be added to twice the angle BAC but to be taken from it 9 If a straight line falling upon the circumference of a circle be produced till it reach the Semidiameter and that part of it which is intercepted between the circumference and the Semidiameter be equal to that part of the Semidiameter which is between the point of concourse and the center the reflected Line will be parallel to the Semidiameter Let any Line AB in the 9th figure be the Semidiameter of the circle whose center is A and upon the circumference BD let the straight Line CD fall and be produced till it cut AB in E so that ED and EA may be equal from the incident Line CD let the Line DF be reflected I say AB and DF will be parallel Let AG be drawn through the point D. Seeing therefore ED and EA are equal the angles EDA and EAD will also be equal But the angles FDG and EDA are equal for each of them is half the angle EDH or FDC Wherefore the angles FDG and EAD are equal and consequently DF and AB are parallel which was to be proved Corollahy If EA be greater then ED then DF and AB being produced will concurre but if EA be less then ED then BA and DH being produced will concurre 10 If from a point within a circle two straight Lines be drawn to the Circumference and their reflected Lines meet in the Circumference of the same circle the angle made by the Lines of Reflection will be a third part of the angle made by the Lines of Incidence From the point B in the 10th figure taken within the circle whose center is A let the two straight lines BC and BD be drawn to the circumference and let their reflected Lines CE and DE meet in the circumference of the same circle at the point E. I say the angle CED will be a third part of the angle CBD Let AC and AD be drawn Seeing therefore the angles CED and CBD together taken are equal to twice the angle CAD as has been demonstrated in the 5th article and the angle CAD twice taken is quadruple to the angle CED the angles CED and CBD together taken will also be equal to the angle CED four times taken and therefore if the angle CED be taken away on both sides there will remain the angle CBD on one side equal to the angle CED thrice taken on the other side which was to be demonstrated Coroll Therefore a point being given within a Circle there may be drawn two Lines from it to the Circumference so as their reflected Lines may meet in the Circumference For it is but trisecting the Angle CBD which how it may be done shall be shewn in the following Chapter CHAP. XX. Of the Dimension of a Circle and the Division of Angles or Arches 1 The Dimension of a Circle neer determined in Numbers by Archimedes and others 2 The first attempt for the finding out of the Dimension of a Circle by Lines 3 The second attempt for the finding out of the Dimension of a Circle from the consideration of the nature of Crookedness 4 The third attempt and some things propounded to be further searched into 5 The Equation of the Spiral of Archimedes with a straight Line 6 Of the Analysis of Geometricians by the Powers of Lines 1 IN the comparing of an Arch of a Circle with a Straight Line many and great Geometricians even from the most ancient times have exercised their wits and more had done the same if they had not seen their pains though undertaken for the common good if not brought to perfection vilified by those that envy the prayses of other men Amongst those Ancient Writers whose Works are come to our hands Archimedes was the first that brought the Length of the Perimeter of a Circle within the limits of Numbers very litle differing from the truth demonstrating the same to be less then three Diameters and a seventh part but greater then three Diameters and ten seventy one parts of the Diameter So that supposing the Radius to consist of 10000000 equal parts the Arch of a Quadrant will be between 15714285 and 15 04225 of the same parts In our times Ludovicus Van Cullen Willebrordus Snellius with joint endeavour have come yet neerer to the truth and pronounced from true Principles that the Arch of a Quadrant putting as before 10000000 for Radius differs not one whole Unity from the number 15707963 which if they had exhibited their arithmetical operations and no man had discovered any errour in that long work of theirs had been demonstrated by them This is the furthest progress that has been made by the way of Numbers and they that have proceeded thus far deserve the praise of Industry Nevertheless if we consider the benefit which is the scope at which all Speculation should aime the improvement they have made has been little or none For any ordinary man may much sooner more accurately find a Straight Line equal to the Perimeter of a Circle and consequently square the Circle by winding a small thred about a given Cylinder then any Geometrician shall do the same by dividing the Radius into 10000000 equal parts But though the length of the Circumference were exactly set out either by Numbers or mechanically or onely by chance yet this would contribute no help at all towards the Section of Angles unless happily these two Problemes To divide a given Angle according to any proportion assigned and To finde a
together taken so that the part next the Vertex be triple to the other part or to the whole straight line as 3 to 4. For let A B C in the 9th fig. be the Sector of a Sphere whose Vertex is the ceâ—ter of the Sphere A whose Axis is A D and the circle upon B C is the common base of the portion of the Sphere and of the Cone whose Vertex is A the Axis of which portion is E D and the halfe thereof F D and the Axis of the Cone A E. Lastly let A G be ¾ of the straight line A F. I say G is the center of Equiponderation of the Sector A B C. Let the straight line F H be drawne of any length making right angles with A F at F and drawing the straight line A H let the triangle A F H be made Then upon the same center A let any arch I K be drawne cutting A D in L and its chord cutting A D in M and dividing M L equally in N let N O be drawne parallel to the straight line F H and meeting with the straight line A H in O. Seeing now B D C is the Spherical Superficies of the portion cut off with a plain passing through B C and cutting the Axis at right angles and seeing F H divides E D the Axis of the portion into two equal parts in F the center of Equiponderation of the Superficies B D C will be in F by the 8th article and for the same reason the center of Equiponderation of the Superficies I L K K being in the straight line A C will be in N. And in like manner if there were drawne between the center of the Sphere A and the outermost Spherical Superficies of the Sector arches infinite in number the centers of Equiponderation of the Sphericall Superficies in which those arches are would be found to be in that part of the Axis which is intercepted between the Superficies it selfe and a plaine passing along by the chord of the arch and cutting the Axis in the middle at right angles Let it now be supposed that the moment of the outermost sphericall Superficies B D C is F H. Seeing therefore the Superficies B D C is to the Superficies I L K in proportion duplicate to that of the arch B D C to the arch I L K that is of B E to I M that is of F H to N O let it be as F H to N O so N O to another N P and again as N O to N P so N P to another N Q and let this be done in all the straight lines parallel to the base F H that can possibly be drawn between the base and the vertex of the triangle A F H. If then through all the points Q there be drawn the crooked line A Q H the figure A F H Q A will be the complement of the first three-siâ—ed figure of two Meanes and the same will also be the moment of all the Sphericall Superficies of which the Solid Sector A B C D is compounded and by consequent the moment of the Sector it selfe Let now F H be understood to be the semidiameter of the base of a right Cone whose side is A H and Axis A F. Wherfore seeing the bases of the Cones which passe through F and N and the rest of the points of the Axis are in proportion duplicate to that of the straight lines F H and N O c. the moment of all the bases together that is of the whole Cone will be the figure it self A F H Q A and therefore the center of Equiponderation of the Cone A F H is the same with that of the solid Sector Wherefore seeing A G is ¾ of the Axis A F the center of Equiponderation of the Cone A F H is in G and therefore the center of the solid Sector is in G also and divides the part A F of the Axis so that A G is triple to G F that is A G is to A F as 3 to 4 which was to be demonstrated Note that when the Sector is a Hemisphere the Axis of the Cone vanisheth into that point which is the center of the Sphere and therefore it addeth nothing to half the Axis of the portion Wherefore if in the Axis of the Hemisphere there be taken from the center ¾ of halfe the Axis that is 3 â— of the Semidiameter of the Sphere there will be the center of Equiponderation of the Hemisphere CHAP. XXIV Of Refraction and Reflection 1 Definitions 2 In perpendicular Motion there is no Refraction 3 Things thrown out of a thinner into a thicker Medium are so refracted that the Angle Refracted is greater then the Angle of Inclination 4 Endeavour which from one point tendeth every way will be so Refracted at that the sine of the Angle Refracted will be to the sine of the Angle of Inclination as the Density of the first Medium is to the Density of the second Medium reciprocally taken 5 The sine of the Refracted Angle in one Inclination is to the sine of the Refracted Angle in another Inclination as the sine of the Angle of that Inclination is to the sine of the Angle of this Inclination 6 If two lines of Incidence having equal Inclination be the one in a thinner the other in a thicker Medium the sine of the angle of Inclination will be a Mean proportional between the two sines of the Refracted angles 7 If the angle of Inclination be semirect and the line of Inclination be in the thicker Medium and the proportion of their Densities be the same with that of the Diagonal to the side of a Square and the separating Superficies be plain the Refracted line will be in the separating Superficies 8 If a Body be carried in a straight line upon another Body and do not penetrate the same but be reflected from it the angle of Reflexion will be equal to the Angle of Incidence 9 The same happens in the generation of Motion in the line of Incidence 1 Definitions 1 REFRACTION is the breaking of that straight Line in which a Body is moved or its Action would proceed in one and the same Medium into two straight lines by reason of the different natures of the two Mediums 2 The former of these is called the Line of Incidence the later the Refracted Line 3 The Point of Refraction is the common point of the Line of Incidence and of the Refracted Line 4 The Refracting Superficies which also is the Separating Superficies of the two Mediums is that in which is the point of Refraction 5 The Angle Refracted is that which the Refracted Line makes in the point of Refraction with that Line which from the same point is drawn perpendicular to the separating Superficies in a different Medium 6 The Angle of Refraction is that which the Refracted line makes with the Line of Incidence produced 7 The Angle of Inclination is
the same Quality as Every Man is a Living Creature Some Man is a Living Creature or No Man is Wise Some Man is not Wise. Of these iâ— the Universal be true the Particular will be true also Contrary are Universal Propositions of different Quality as Every Man is happy No Man is happy And of these if one be true the other is false also they may both be false as in the example given Subcontrary are Particular Propositions of different Quality as Some Man is learned Some Man is not learned which cannot be both false but they may be both true Contradictory are those that differ both in Quantity and Quality as Every Man is a Living Creature Some Man is not a Living Creature which can neither be both true nor both false 18 A Proposition is said to follow from two other Propositions when these being granted to be true it cannot be denyed but the other is true also For example let these two Propositions Every Man is a Living Creature and Every Living Creature is a Body be supposed true that is that Body is the Name of Every Living Creature and Living Creature the Name of Every Man Seeing therefore if these be understood to be true it cannot be understood that Body is not the name of Every man that is that Every Man is a Body is false this Proposition will be said to follow from those two or to be necessarily inferred from them 19 That a true Proposition may follow from false Propositions may happen sometimes but false from true never For if these Every Man is a Stone and Every Stone is a Living Creature which are both false be granted to be true it is granted also that Living Creature is the name of Every Stone and Stone of Every Man that is that Living Creature is the Name of Every Man that is to say this Proposition Every Man is a Living Creature is true as it is indeed true Wherefore a true Proposition may sometimes follow from false But if any two Propositions be true a false one can never follow from them For if true follow from false for this reason onely that the false are granted to be true then truth from two truths granted will follow in the same manner 20 Now seeing none but a true Proposition will follow from true and that the understanding of two Propositions to be true is the cause of understanding that also to be true which is deduced from them the two Antecedent Propositions are commonly called the Causes of the inferred Proposition or Conclusion And from hence it is that Logicians say the Premisses are Causes of the Conclusion which may passe though it be not properly spoken for though Understanding be the cause of Understanding yet Speech is not the cause of Speech But when they say the Cause of the Properties of any thing is the Thing it self they speake absurdly Eor example if a Figure be propounded which is Triangular Seeing every Triangle has all its angles together equal to two right angles from whence it follows that all the angles of that Figure are equal to two right angles they say for this reason that that Figure is the Cause of that Equality But seeing the Figure does not it self make its angles and therefore cannot be said to be the Efficient-Cause they call it the Formall-Cause whereas in deed it is no Cause at all nor does the Property of any Figure follow the Figure but has its Being at the same time with it only the Knowledge of the Figure goes before the Knowledge of the Properties and one Knowledge is truly the Cause of another Knowledge namely the Efficient-Cause And thus much concerning Proposition which in the Progress of Philosophy is the first Step like the moving towards of one Foot By the due addition of another Step I shall proceed to Syllogisme and make a compleat Pace Of which in the next Chapter CHAP. IV. Of Syllogisme 1 The Definition of Syllogisme 2 In a Syllogisme there are but three Termes 3 Major Minor and Middle Term also Major and Minor Proposition what they are 4 The Middle Terme in every Syllogisme ought to be determined in both the Propositions to one and the same thing 5 From two Particular Propositions nothing can be concluded 6 A Syllogisme is the Collection of two Propositions into one Summe 7 The Figure of a Syllogisme what it is 8 What is in the mind answering to a Syllogisme 9 The first Indirect Figure how it is made 10 The second Indirect Figure how made 11 How the third Indirect Figure is made 12 There are many Moods in every Figure but most of them Uselesse in Philosophy 13 An Hypotheticall Syllogisme when equipollent to a Categoricall 1. A Speech consisting of three Propositions from two of which the third followes is called a SYLLOGISME and that which followes is called the Conclusion the other two Premisses For example this Speech Every man is a Living Creature Every Living Creature is a Body therefore Every Man is a Body is a Syllogisme because the third Proposition follows from the two first that is if those be granted to be true this must also be granted to be true 2 From two Propositions which have not one Terme common no Conclusion can follow and therefore no Syllogisme can be made of them For let any two Premisses A man is a Living Creature A Tree is a Plant be both of them true yet because it cannot be collected from them that Plant is the Name of a Man or Man the Name of a Plant it is not necessary that this Conclusion A Man is a Plant should be true Corollary Therefore in the Premisses of a Syllogisme there can be but three Termes Besides there can be no Terme in the Conclusion which was not in the Premisses For let any two Premisses be A Man is a Living Creature A Living Creature is a Body yet if any other Terme be put in the Conclusion as Man is two footed though it be true it cannot follow from the Premisses because from them it cannot be collected that the Name Two footed belongs to a Man and therefore againe In every Syllogisme there can be but three Termes 3 Of these Termes that which is the Predicate in the Conclusion is commonly called the Major that which is the Subject in the Conclusion the Minor and the other is the Middle Term as in this Syllogisme A Man is a Living Creature A Living Creature is a Body therefore A Man is a Body Body is the Major Man the Minor and Living Creature the Middle Term. Also of the Premisses that in which the Major Terme is found is called the Major Proposition and that which has the Minor Term the Minor Proposition 4 If the Middle Terme be not in both the Premisses determined to one and the same singular thing no Conclusion will follow nor Syllogisme be made For let the Minor Terme be Man the Middle Terme Living Creature and the Major Term
and the Properties of Straight Parallels 13 The Circumferences of Circles are to one another as their Diameters are 14 In Triangles Straight Lines parallel to the Bases are to one another as the parts of the Sides which they cut off from the Vertex 15 By what Fraction of a Straight Line the Circumference of a Circle is made 16 That an Angle of Contingence is Quantity but of a Different kinde from that of an Angle simply so called and that it can neither add nor take away any thing from the same 17 That the Inclination of Plains is Angle simply so called 18 A Solid Angle what it is 19 What is the Nature of Asymptotes 20 Situation by what it is determined 21 What is like Situation What is Figure and what are like Figures 1 BEtween two points given the shortest Line is that whose extreme points cannot be drawn further asunder withour altering the quantity that is without altering the proportion of that line to any other line given For the Magnitude of a Line is computed by the greatest distance which may be between its extreme points So that any one Line whether it be extended or bowed has alwayes one and the same Length because it can have but one greatest distance between its extreme points And seeing the action by which a Straight Line is made Crooked or contrarily a Crooked Line is made Straight is nothing but the bringing of its extreme points neerer to one another or the setting of them further asunder a CROOKED Line may rightly be defined to be That whose extreme points may be understood to be drawn further asunder and a STRAIGHT Line to be That whose extreme points cannot be drawn further asunder and comparatively A more Crooked to be That line whose extreme points are neerer to one another then those of the other supposing both the Lines to be of equal Length Now howsoever a Line be bowed it makes alwayes a Sinus or Cavity sometimes on one side sometimes on another So that the same Crooked Line may either have its whole Cavity on one Side onely or it may have it part on one side and part on other sides Which being well understood it will be easie to understand the following Comparisons of Straight and Crooked Lines First If a Straight a Crooked Line have their Extreme points common the Crooked Line is longer then the Straight Line For if the extreme points of the Crooked Line be drawn out to their greatest distance it wil be made a straight line of which that which was a Straight Line from the beginning will be but a part and therefore the Straight Line was shorter then the Crooked Line which had the same extreme points And for the same reason if two Crooked Lines have their extreme points common and both of them have all their cavity on one and the same side the outermost of the two will be the longest Line Secondly A Straight Line and a perpetually Crooked Line cānot be coincident no not in the least part For if they should then not onely some Straight Line would have its extreme points common with some Crooked Line but also they would by reason of their coincidence be equal to one another which as I have newly shewn cannot be Thirdly Between two points given there can be understood but one straight Line because there cannot be more then one least Interval or Length between the same points For if there may be two they will either be coincident and so both of them will be one Straight Line or if they be not coincident then the application of one to the other by extension will make the extended Line have its extreme points at greater distance then the other and consequently it was Crooked from the beginning Fourthly From this last it follows that two Straight Lines cannot include a Superficies For if they have both their extreme points common they are coincident and if they have but one or neither of them common then at one or both ends the extreme points will be disjoyned and include no Superficies but leave all open and undetermined Fifthly Every part of a Straight Line is a Straight Line For seeing every part of a Straight Line is the least that can be drawn between its own extreme points if all the parts should not constitute a Straight Line they would all together be longer then the whole Line 2 APLAIN or a Plain Superficies is that which is described by a Straight Line so moved that all the several points thereof describe several Straight Lines A straight line therefore is necessarily all of it in the same Plain which it describes Also the Straight Lines which are made by the points that describe a Plain are all of them in the Same Plain Moreover if any Line whatsoever be moved in a Plain the Lines which are described by it are all of them in the same Plain All other Superficies which are not Plain are Crooked that is are either Concave or Convex And the same Comparisons which were made of Straight and Crooked Lines may also be made of Plain and Crooked Superficies For First If a Plain and a Crooked Superficies be terminated with the same Lines the Crooked Superficies is greater then the Plain Superficies For if the Lines of which the Crooked Superficies consists be extended they will be found to be longer then those of which the Plain Superficies consists which cannot be extended because they are Straight Secondly Two Superficies wherof the one is Plain and the other continually Crooked cannot be coincident no not in the least part For if they were coincident they would be equal nay the same Superficies would be both Plain and Crooked which is impossible Thirdly Within the same terminating Lines there can be no more then one Plain Superficies because there can be but one least Superficies within the same Fourthly No number of Plain Superficies can include a Solid unless more then two of them end in a Common Vertex For if two Plains have both the same terminating Lines they are coincident that is they are but one Superficies and if their terminating Lines be not the same they leave one or more sides open Fifthly Every part of a Plain Superficies is a Plain Superficies For seeing the whole Plain Superficies is the least of all those that have the same terminating Lines and also every part of the same Superficies is the least of all those that are terminated with the same Lines if every part should not constitute a Plain Superficies all the parts put together would not be equal to the whole 3 Of Straightness whether it be in Lines or in Superficies there is but one kinde but of Crookedness there are many kindes for of Crooked Magnitudes some are Congruous that is are coincident when they are applyed to one another others are Incongruous Again some are 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 or Uniform that is have their parts howsoever taken congruous to one
another others are 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 or of several Forms Moreover of such as are Crooked some are Continually Crooked others have parts which are not Crooked 4 If a Straight Line be moved in a Plain in such manner that while one end of it stands still the whole Line be carried round about til it come again into the same place from whence it was first moved it will describe a plain Superficies which will be terminated every way by that Crooked Line which is made by that end of the Straight Line which was carried round Now this Superficies is called a CIRCLE and of this Circle the Unmoved Point is the the Center the Crooked Line which terminates it the Perimeter and every part of that Crooked Line a Circumference or Arch the straight Line which generated the Circle is the Semidiameter or Radius and any straight Line which passeth through the Center and is terminated on both sides in the Circumference is called the Diameter Moreover every point of the Radius which describes the Circle describes in the same time it s own Perimeter terminating its own Circle which is said to be Concentrick to all the other Circles because this and all those have one common Center Wherefore in every Circle all Straight Lines from the Center to the Circumference are equal For they are all coincident with the Radius which generates the Circle Also the Diameter divides both the Perimeter and the Circle it self into two equal parts For if those two parts be applyed to one another and the Semiperimeters be coincident then seeing they have one common Diameter they will be equal and the Semicircles will be equal also for these also will be coincident But if the Semiperimeters be not coincident then some one straight Line which passes through the Center which Center is in the Diameter will be cut by them in two points Wherefore seeing all the straight Lines from the Center to the Circumference are equal a part of the same straight Line will be equal to the whole which is impossible For the same reason the Perimeter of a Circle will be Uniform that is any one part of it will be coincident with any other equal part of the same 5 From hence may be collected this property of a Straight Line namely that it is all conteined in that Plain which conteins both its extreme points For seeing both its extreme points are in the Plain that Straight Line which describes the Plain will pass through them both and if one of them be made a Center and at the distance between both a Circumference be described whose Radius is the Straight Line which describes the Plain that Circumference will pass through the other point Wherefore between the two propounded points there is one straight line by the Definition of a Circle conteined wholly in the propounded Plain and therefore if another straight Line might be drawn between the same points and yet not be conteined in the same Plain it would follow that between two points two straight lines may be drawn which has been demonstrated to be impossible It may also be collected That if two Plains cut one another their common section will be a straight Line For the two extreme points of the intersection are in both the intersecting Plains and between those points a straight Line may be drawn but a straight Line between any two points is in the same Plain in which the Points are and seeing these are in both the Plains the straight line which connects them will also be in both the same Plains and therefore it is the coÌ„mon section of both And every other Line that can be drawn between those points will be either coincident with that Line that is it will be the Same Line or it will not be coincident and then it wil be in neither or but in one of those Plains As a straight Line may be understood to be moved round about whilest one end thereof remains fixed as the Center so in like manner it is easie to understand that a Plain may be circumduced about a straight line whilest the straight line remaines still in one and the same place as the Axis of that motion Now from hence it is manifest that any three Points are in some one Plain For as any two Points if they be connected by a straight Line are understood to be in the same Plaine in which the straight Line is so if that Plaine be circumduced about the same straight Line it will in its revolution take in any third Point howsoever it be situate and then the three Points will be all in that Plaine and consequently the three straight Lines which connect those Points will also be in the same Plain 6 Two Lines are said to Touch one another which being both drawne to one and the same point will not cut one another though they be produced produced I say in the same manner in which they were generated And therefore if two straight Lines touch one another in any one point they wil be contiguous through their whole length Also two Lines continually crooked wil do the same if they be congruous and be applyed to one another according to their congruity otherwise if they be incongruously applyed they will as all other crooked Lines touch one another where they touch but in one point onely Which is manifest from this that there can be no congruity between a straight line and a line that is continually crooked for otherwise the same line might be both straight and crooked Besides when a straight line touches a crooked line if the straight line be never so little moved about upon the point of contact it will cut the crooked line for seeing it touches it but in one point if it incline any way it will do more then touch it that is it will either be congruous to it or it will cut it but it cannot be congruous to it and therefore it will cut it 7 An Angle according to the most general acception of the word may be thus defined When two Lines or many Superficies concurre in one sole point and diverge every where else the quantity of that divergence is an ANGLE And an Angle is of two â—orts for first it may be made by the concurrence of Lines and then it is a Superficiall Angle or by the concurrence of Superficies and then it is called a Solid Angle Again from the two wayes by which two lines may diverge from one another Superficial Angles are divided into two kindes For two straight lines which are applyed to one another and are contiguous in their whole length may be separated or pulied open in such manner that their concurrence in one point will still remain And this Separation or Opening may be either by Circular Motion the Center whereof is their point of concurrence and the Lines will still retâ—in their straightness the quantity of which Separation or Divergence is an Angle
can it be greater because if any straight line whatsoever lesse then B T be drawâ— below B T parallel to it and terminated in the straight lineâ— X B and X T it would cut the arch B F and so the Sine of some one of the parts of the arch B F taken so often as that small arch is found in the whole arch B F would be greater then so many of the same arches which is absurd Wherefore the Straight line B T is equal to the Arch B F the Straight line B V equal to the Arch of the Quadrant B F D and B V four times taken equal to the Perimeter of the Circle described with the Radius A B. Also the Arch B F and the Straight line B T are every where divided into the same proportions and consequently any given Angle whether greater or less then B A F may be divided into any proportion given But the straight line B V though its magnitude fall within the terms assigned by Archimedes is found if computed by the Canon of Sines to be somwhat greater then that w ch is exhibited by the Ludolphine numbers Nevertheless if in the place of B T another straight line though never so little less be substituted the division of Angles is immediatly lost as may by any man be demonstrated by this very Scheme Howsoever if any man think this my Straight line B V to be too great yet seeing the Arch and all the Parallels are every where so exactly divided and B V comes so neer to the truth I desire he would seach out the reason Why granting B V to be precisely true the Arches cut off should not be equal But some man may yet ask the reason why the straight lines drawn from X through the equal parts of the arch B F should cut off in the Tangent B V so many straight lines equal to them seeing the connected straight line X V passes not through the point D but cuts the straight line A D produced in l and consequently require some determination of this Probleme Concerning which I will say what I think to be the reason namely that whilest the magnitude of the Arch doth not exceed the magnitude of the Radius that is the magnitude of the Tangent B C both the Arch and the Tangent are cut alike by the straight lines drawn from X otherwise not For A V being connected cutting the arch B H D in I if X C being drawn should cut the same arch in the same point I it would be as true that the Arch B I is equal to the Radius B C as it is true that the Arch B F is equal to the straight line B T and drawing X K it would cut the arch B I in the midst in i Also drawing A i and producing it to the Tangent B C in k the straight line B k will be the Tangent of the arch B i which arch is equal to half the Radius and the same straight line B k will be equal to the straight line k I. I say all this is true if the preceding demonstration be true and consequently the proportional section of the Arch and its Tangent proceeds hitherto But it is manifest by the Golden Rule that taking B h double to B T the line X h shall not cut off the arch B E which is double to the arch B F but a much greater For the magnitude of the straight lines X M X B and M E being known in numbers the magnitude of the straight line cut off in the Tangent by the straight line X E produced to the Tangent may also be known and it will be found to be less then B h Wherfore the straight line Xh being drawn will cut off a part of the arch of the Quadrant greater then the arch B E. But I shall speak more fully in the next Article concerning the magnitude of the arch B I. And let this be the first attempt for the finding out of the dimension of a Circle by the Section of the arch B F. 3 I shall now attempt the same by arguments drawn from the nature of the Crookedness of the Circle it self but I shall first set down some Premisses necessary for this speculation and First If a Straight line be bowed into an Arch of a Circle equal to it as when a stretched thred which toucheth a Right Cylinder is so bowed in every point that it be every where coincident with the Perimeter of the base of the Cylinder the Flexion of that line will be equal in all its points and consequently the Crookedness of the Arch of a Circle is every where Uniform which needs no other demonstration then this That the Perimeter of a Circle is an Uniform line Secondly and consequently If two unequal Arches of the same Circle be made by the bowing of two straight lines equal to them the Flexion of the longer line whilest it is bowed into the greater Arch is greater then the Flexion of the shorter line whilest it is bowed into the lesser Arch according to the proportion of the Arches themselves and consequently the Crookedness of the greater Arch is to the Crookedness of the lesser Arch as the greater Arch is to the lesser Arch. Thirdly If two unequal Circles and a straight line touch one another in the same point the Crookedness of any Arch taken in the lesser Circle will be greater then the Crookedness of an Arch equal to it taken in the greater Circle in reciprocal proportion to that of the Radii with which the Circles are described or which is all one any straight line being drawn from the point of Contact till it cut both the circumferences as the part of that straight line cut off by the circumference of the greater Circle to that part which is cut off by the circumference of the lesser Circle For let A B and A C in the second figure be two Circles touching one another and the straight line A D in the point A and let their Centers be E and F and let it be supposed that as A E is to A F so is the Arch A B to the Arch A H. I say the Crookedness of the Arch A C is to the Crookedness of the Arch A H as A E is to A F. For let the straight line A D be supposed to be equal to the Arch A B and the straight line A G to the Arch A C and let A D for example be double to A G. Therefore by reason of the likeness of the Arches A B and A C the straight line A B will be double to the straight line A C and the Radius A E double to the Radius A F and the Arch A B double to the Arch A H. And because the straight line A D is so bowed to be coincident with the Arch A B equal to it as the straight line A G is bowed to be coincident with the Arch A C equal
equal that no inequality can be discovered between them I will therefore leave this to be further searched into For though it be almost out of doubt that the Straight Line BP and the arch BMD are equal yet that may not be received without demonstration and means of Demonstration the Circular Line admitteth none that is not grounded upon the nature of Flexion or of Angles But by that way I have already exhibited a Straight Line equal to the Arch of a Quadrant in the First and Second aggression It remains that I prove DT to be equal to the Sine of 45 degrees In BA produced let AV he taken equal to the Sine of 45 degrees and drawing and producing VH it will cut the arch of the Quadrant CNA in the midst in N and the same arch again in O and the Straight line DC in T so that DT will be equal to the Sine of 45 degrees or to the straight line AV also the Straight line VH will be equal to the straight line HI or the Sine of 60 degrees For the square of AV is equal to two squares of the Semiradius and consequently the square of VH is equal to three Squares of the Semiradius But HI is a mean proportional between the Semiradius and three Semiradii and therefore the square of HI is equal to three Squares of the Semiradius Wherefore HI is eqval to HV But because AD is cut in the midst in H therefore VH and HT are equal and therefore also DT is equal to the Sine of 45 degrees In the Radius BA let BX be taken equal to the Sine of 45 degrees for so VX will be equal to the Radius and it will be as VA to AH the Semiradius so VX the Radius to XN the Sine of 45 degrees Wherefore VH produced passes through N. Lastly upon the center V with the Radius VA let the arch of a circle be drawn cutting VH in Y which being done VY will be equal to HO for HO is by construction equal to the Sine of 45 degrees and YH will be equal to OT therefore VT passes through O. All which was to be demonstrated I will here add certain Problemes of which if any Analyst can make the construction he will thereby be able to judge clearly of what I have now said concerning the dimension of a Circle Now these Problems are nothing else at least to sense but certain symptomes accompanying the construction of the first and third figure of this Chapter Describing therefore again the Square ABCD in the fifth figure and the three Quadrants ABD BCA and DAC let the Diagonals AC BD be drawn cutting the arches BHD CIA in the middle in H and I the straight lines EF and GL dividing the square ABCD into four equal squares and trisecting the arches BHD and CIA namely BHD in K and M and CIA in M and O. Then dividing the arch BK in the midst in P let QP the Sine of the arch BP be drawn and produced to R so that QR be double to QP and connecting KR let it be produced one way to BC in S and the other way to BA produced in T. Also let BV be made triple to BS and consequently by the second article of this Chapter equall to the arch BD. This construction is the same with that of the first figure which I thought fit to renew discharged of all lines but such as are necessary for my present purpose In the first place therefore if AV be drawn cutting the arch BHD in X and the side DC in Z I desire some Analyst would if he can give a reason Why the straight lines TE and TC should cut the arch BD the one in Y the other in X so as to make the arch BY equal to the arch YX or if they be not equal that he would determine their difference Secondly if in the side DA the straight line Da be taken equal to DZ and Va be drawn Why Va and VB should be equal or if they be not equal What is the difference Thirdly drawing Zb parallel and equal to the side CB cutting the arch BHD in c and drawing the straight line Ac and producing it to BV in d Why Ad should be equal and parallel to the straight line aV and consequently equal also to the arch BD. Fourthly drawing eK the Sine of the arch BK taking in eA produced ef equal to the Diagonal AC and connecting fC Why fC should pass through a which point being given the length of the arch BHD is also given and c and why fe and fc should be equal or if not why unequal Fifthly drawing fZ I desire he would shew Why it is equal to BV or to the arch BD or if they be not equal What is their difference Sixtly granting fZ to be equal to the arch BD I desire he would determine whether it fall all without the arch BCA or cut the same or touch it and in what point Seventhly the Semicircle BDg being completed Why gI being drawn and produced should pass through X by which point X the length of the arch BD is determined And the same gI being yet further produced to DC in h Why Ad which is equal to the arch BD should pass through that point h. Eighthly upon the Center of the square ABCD which let be k the arch of the quadrant EiL being drawn cutting eK produced in i Why the drawn straight line iX should be parallel to the side CD Ninthly in the sides BA and BC taking Bl and Bm severally equal to half BV or to the arch BH and drawing mn parallel and equal to the side BA cutting the arch BD in o Why the straight line wich connects Vl should pass through the point o Tenthly I would know of him Why the straight line which connects aH should be equal to Bm or if not how much it differs from it The Analyst that can solve these Problemes without knowing first the length of the arch BD or using any other known Method then that which proceeds by perpetual bisection of an angle or is drawn from the consideration of the nature of Flexion shall do more then ordinary Geometry is able to perform But if the Dimension of a Circle cannot be found by any other Method then I have either found it or it is not at all to be found From the known Length of the Arch of a Quadrant and from the proportional Division of the Arch and of the Tangent BC may be deduced the Section of an Angle into any given proportion as also the Squaring of the Circle the Squaring of a given Sector and many the like propositions which it is not necessary here to demonstrate I will therefore onely exhibit a Straight line equal to the Spiral of Archimedes and so dismiss this speculation 5 The length of the Perimeter of a Circle being found that Straight line is also found which
weakned by little and little But this cannot be done but by the long continuance of action or by actions often repeated and therefore Custome begets that Facicility which is commonly and rightly called Habit and it may be defined thus HABIT is Motion made more easie and ready by Custome that is to say by perpetual endeavour or by iterated endevours in a way differing from that in which the Motion proceeded from the beginning and opposing such endeavours as resist And to make this more perspicuous by example We may observe that when one that has no skill in Musique first puts his hand to an Instrument he cannot after the first stroke carry to his hand to the place where he would make the second stroke without taking it back by a new endeavour and as it were beginning again pass from the first to the second Nor will he be able to go on to the third place without another new endeavour but he will be forced to draw back his hand again and so successively by renewing his endeavour at every stroke till at the last by doing this often and by compounding many interrupted motions or endeavours into one equal endeavour he be able to make his hand go readily on from stroke to stroke in that order and way which was at the first designed Nor are Habits to be observed in living creatures only but also in Bodies inanimate For we find that when the Lath of a Crossbow is strongly bent and would if the impediment were removed return again with great force if it remain a long time bent it will get such a Habit that when it is loosed and left to its own freedome it will not onely not restore it self but will require as much force for the bringing of it back to its first posture as it did for the bending of it at the first CHAP. XXIII Of the Center of Equiponderation of Bodies pressing doâ—â—ards in straight Parallel Lines 1 Definitions and Suppositions 2 Two Plains of Equiponderation are nâ—â— parallel 3 The Center of Equiponderation is in every Plain of Equiponderation 4 The Moments of equal Ponderants are to one another as their distances from the center of the Scale 5,6 The Moments of unequal Ponderants have their proportion to one another compounded of the proportions of their Waights and distances from the center of the Scale reciprocally taken 7. If two Ponderants have their Moments and Distances from the Center of the Scale in reciprocal proportion they are equally poised and contrarily 8 If the parts of any Ponderant press the Beam of the Scale every where equally all the parts cut out off reckoned from the Center of the Scale will have their Moments in the same proportion with that of the parts of a Triangle cut off from the Vertex by straight Lines parallel to the base 9 The Diameter of Equiponderation of Figures which are deficient according to commensurable proportions of their altitudes and bases divides the Axis so that the part taken next the vertex is to the other part as the complete figure to the deficient figure 10 The diameter of Equiponderation of the Complement of the half of any of the said deficient figures divides that line which is drawn through the vertex parallel to the base so that the part next the vertex is to the other part as the complete figure to the Complement 11 The Center of Equiponderation of the half of any of the desicient figures in the first row of the Table of the 3d. Article of the 17th Chapter may be found out by the numbers of the second row 12 The center of Equiponderation of the half of any of the figures in the second row of the same Table may be found out by the numbers of the fourth row 13 The Center of Equiponderation of the half of any of the figures in the same Table being known the Center of the Excess of the same figure above a Triangle of the same altitude and base is also known 14 The Center of Equiponderation of a solid Sector is in the Axis so divided that the part next the Vertex be to the whole Axis want half the Axis of the portion of the Sphere as 3 to 4. 1 Definitions 1 A Scale is a straight line whose middle point is immoveable all the rest of its points being at liberty and that part of the Scale which reaches from the center to either of the waights is called the Beam 2 Equiponderation is when the endeavour of one Body which presses one of the Beams resists the endeavour of another Body pressing the other Beam so that neither of them is moved and the Bodies when neither of them is moved are said to be Equally poised 3 Waight is the aggregate of all the Endeavours by which all the points of that Body which presses the Beam tend downwards in lines parallel to one another and the Body which presses is called the Ponderant 4 Moment is the Power which the Ponderant has to move the Beam by reason of a determined situation 5 The plain of Equiponderation is that by which the Ponderant is so divided that the Moments on both sides remain equal 6 The Diameter of Equiponderation is the common Section of the two Plains of Equiponderation and is in the straight line by which the waight is hanged 7 The Center of Equiponderation is the common point of the two Diameters of Equiponderation Suppositions 1 When two Bodies are equally pois'd if waight be added to one of them and not to the other their Equiponderation ceases 2 When two Ponderants of equal magnitude and of the same Species or matter press the Beam on both sides at equal distances from the center of the Scale their Moments are equal Also when two Bodies endeavour at equal distances from the center of the Scale if they be of equal magnitude and of the same Species their Moments are equal 2 No two Plains of Equiponderation are parallel Let A B C D in the first figure be any Ponderant whatsoever and in it let E F be a Plain of Equiponderation parallel to which let any other Plain be drawn as G H. I say G H is not a Plain of Equiponderation For seeing the parts A E F D and E B C F of the Ponderant A B C D are equally pois'd and the weight E G H F is added to the part A E F D and nothing is added to the part E B C F but the weight E G H F is taken from it therefore by the first Supposition the parts A G H D and G B C H will not be equally pois'd and consequently G H is not a Plain of Equiponderation Wherefore No two Plains of Equiponderation c. Which was to be proved 3 The Center of Equiponderation is in every Plain of Equiponderation For if another Plain of Equiponderation be taken it will not by the last Article be parallel to the former Plain and therefore both those Plains will
times of their descents cannot be easily measured with sufficient exactness and also because the places neer the Poles are inaccessible Nevertheless this we know that by how much the neerer we come to the Poles by so much the greater are the Flakes of the Snow that falls and by how much the more swiftly such Bodies descend as are fluid and dissipable by so much the smaller are the particles into which they are dissipated 5 Supposing therefore this to be the cause of the Descent of Heavy Bodies it will follow that their motion will be accelerated in such manner as that the spaces which are transmitted by them in the several times will have to one another the same proportion which the odd numbers have in succession from Unity For if the straight line EA be divided into any number of equal parts the Heavy Body descending will by reason of the perpetual action of the Diurnal motion receive from the aire in every one of those times in every several point of the streight line EA a several new and equal impulsion and therefore also in every one of those times it will acquire a several and equal degree of celerity And from hence it follows by that which Galilaeus hath in his Dialogues of Motion demonstrated that Heavy Bodies descend in the several times with such differences of transmitted spaces as are equal to the differences of the square numbers that succeed one another from Unity which square numbers being 1 4 9 16 c. their differences are 3 5 7 that is to say the odd numbers which succeed another from Unity Against this cause of Gravity which I have given it will perhaps be objected that if a Heavy Body be placed in the bottom of some hollow Cylinder of Iron or Adamant and the bottom be turned upwards the Body will descend though the aire above cannot depress it much less accelerate its motion But it is to be considered that there can be no Cylinder or Cavern but such as is supported by the Earth and being so supported is together with the Earth carried about by its diurnal Motion For by this means the bottom of the Cylinder will be as the Superficies of the Earth and by thrusting off the next and lowest aire will make the uppermost aire depress the Heavy Body which is at the top of the Cylinder in such manner as is above explicated 6 The Gravity of Water being so great as by experience wee find it is the reason is demanded by many why those that Dive how deep soever they go under water do not at all feel the weight of the water which lyes upon them And the cause seems to be this that all Bodies by how much the Heavier they are by so much the greater is the endeavour by which they tend downwards But the Body of a Man is Heavier then so much water as is equal to it in magnitude and therefore the endeavour downwards of a Mans Body is greater then that of water And seeing all endeavour is motion the Body also of a Man will be carried towards the bottom with greater Velocity then so much water Wherefore there is greater Reaction from the bottom and the Endeavour upwards is equal to the endeavour downwards whether the water be pressed by water or by another Body which is Heavier then water And therefore by these two opposite equal endeavours the endeavour both ways in the water is taken away and consequently those that Dive are not at all pressed by it Coroll From hence also it is manifest that water in water hath no Waight at all because all the parts of water both the parts above and the parts that are directly under tend towards the bottom with equal endeavour and in the same straight lines 7 If a Body float upon the water the waight of that Body is equal to the waight of so much water as would fill the place which the immersed part of the Body takes up within the water Let EF in the 3d figure be a Body floating in the water ABCD and let the part E be above and the other part F under the water I say the waight of the whole Body EF is equal to the waight of so much water as the Space F will receive For seeing the waight of the Body EF forceth the water out of the space F and placeth it upon the Superficies AB where it presseth downwards it follows that from the resistance of the bottom there will also be an endeavour upwards And seeing again that by this endeavour of the water upwards the Body EF is lifted up it follows that if the endeavour of the Body downwards be not equal to the endeavour of the water upwards either the whole Body EF will by reason of that inequality of their endeavours or moments be raised out of the water or else it will descend to the bottom But it is supposed to stand so as neither to ascend nor descend Wherefore there is an Aequilibrium between the two endeavours that is to say the waight of the Body EF is equal to the waight of so much water as the Space F will receive Which was to be proved 8 From hence it follows that any Body of how great magnitude soever provided it consist of matter less Heavy then water may nevertheless float upon any quantity of water how little soever Let ABCD in the 4th figure be a vessel and in it let EFGH be a Body consisting of matter which is less Heavy then water and let the space AGCF be filled with water I say the Body EFGH will not sink to the bottom DC For seeing the matter of the Body EFGH is less Heavy then Water if the whole space without ABCD were full of Water yet some part of the Body EFGH as EFIK would be above the Water and the waight of so much water as would fill the space IGHK would be equal to the waight of the whole Body EFGH and consequently GH would not touch the bottom DC As for the sides of the vessel it is no matter whether they be hard or fluid for they serve onely to terminate the Water which may be done as well by water as by any other matter how hard soever and the water without the Vessel is terminated somewhere so as that it can spread no further The part therefore EFIG will be extant above the water AGCF which is contained in the vessel Wherefore the Body EFGH will also float upon the water AGCF how little soever that water be which was to be demonstrated 9 In the 4th Article of the 26th Chapter there is brought for the proving of Vacuum the experiment of water enclosed in a vessel which water the Orifice above being opened is ejected upwards by the impulsion of the aire It is therefore demanded seeing water is Heavier then aire how that can be done Let the 2d figure of the same 26th Chap. be considered where the water is with great force injected by
that the Proportion of the first Antecedent to the first Consequent is the same with that of the second Antecedent to the second Consequent And when four Magnitudes are thus to one another in Geometrical Proportion they are called Proportionals and by some more briefly Analogisme And Greater Proportion is the Proportion of a Greater Antecedent to the same Consequent or of the same Antecâ—dent to a Less Consequent and when the Proportion of the first Antecedent to the first Consequent is greater then that of the second Antecedent to the second Consequent the four Magnitudes which are so to one another may be called Hyperlogisme Less Proportion is the Proportion of a Less Antecedent to the same Consequent or of the same Antecedent to a Greater Consequent and when the Proportion of the first Antecedent to the first Consequent is less then that of the second to the second the four Magnitudes may be called Hypologisme 5 One Arithmetical Proportion is the Same with another Arithmetical Proportion when one of the Antecedents exceeds its Consequent or is exceeded by it as much as the other Antecedent exceeds its Consequent or is exceeded by it And therefore in four Magnitudes Arithmetically Proportional the sum of the Extremes is equal to the sum of the Means For if A. B C. D be Arithmetically Proportional and the Difference on both sides be the same Excess or the same Defect E then B+C if A be greater then B will be equal to A − E+C and A+D will be equal to A+C − E But A − E+C and A+C − E are equal Or if A be less then B then B+C will be equal to A+E+C and A+D will be equal to A+C+E But A+E+C and A+C+E are equal Also if there be never so many Magnitudes Arithmetically Proportional the Sum of them all will be equal to the Product of half the number of the Terms multiplyed by the Sum of the Extremes For if A. B C. D E. F be Arithmetically Proportional the Couples A+F B+E C+D will be equal to one another and their Sum will be equal to A+F multiplyed by the number of their Combinations that is by half the number of the Terms If of four Unequal Magnitudes any two together taken be equal to the other two together taken then the greatest and the least of them will be in the same Combination Let the Unequal Magnitudes be A B C D and let A+B be equal to C+D let A be the greatest of them all I say B will be the least For if it may be let any of the rest as D be the least Seeing therefore A is greater then C and B then D A+B will be greater then C+D which is contrary to what was supposed If there be any four Magnitudes the Sum of the greatest and least the Sum of the Means the difference of the two greatest and the difference of the two least will be Arithmetically Proportional For let there be four Magnitudes whereof A is the greatest D the least and B and C the Means I say A+D B+C A − B. C − D are Arithmetically Proportional For the difference between the first Antecedent and its Consequent is this A+D − B − C and the difference between the second Antecedent and its Consequent this A − B − C+D but these two Differences are equal and therefore by this 5th Article A+D B+C A − B. C − D are Arithmetically Proportional If of four Magnitudes two be equal to the other two they will be in reciprocal Arithmetical Proportion For let A+B be equal to C+D I say A. C D. B are Arithmetically Proportional For if they be not let A. C D. E supposing E to be greater or less then B be Arithmetically Proportional and then A+E will be equal to C+D wherefore A+B and C+D are not equal which is contrary to what was supposed 6 One Geometrical Proportion is the same with another Geometrical Proportion when the same Cause producing equal Effects in equal Times determines both the Proportions If a Point Uniformly moved describe two Lines either with the same or different Velocity all the parts of them which are contemporary that is which are described in the same time will be Two to Two in Geometrical Proportion whether the Antecedents be taken in the same Line or not For from the point A in the 10 Figure at the end of the 14 Chapter let the two Lines A D A G be described with Uniform Motion and let there be taken in them two parts AB AE and again two other parts AC AF in such manner that AB AE be contemporary and likewise AC AF contemporary I say first taking the Antecedents AB AC in the Line AD and the Consequents AE AF in the Line AG that AB AC AE AF are Proportionals For seeing by the 8th Chapter and the 15 Article Velocity is Motion considered as determined by a certain Length or Line in a certain Time transmitted by it the quantity of the Line AB will be determined by the Velocity and Time by which the same AB is described and for the same reason the quantity of the Line AC will be determined by the Velocity and Time by which the same AC is described and therefore the proportion of AB to AC whether it be Proportion of Equality or of Excess or Defect is determined by the Velocities and Times by which AB AC are described But seeing the Motion of the Point A upon AB and AC is Uniform they are both desribed with equal Velocity and therefore whether one of them have to the other the Proportion of Majority or of Minority the sole cause of that Proportion is the difference of their Times and by the same reason it is evident that the proportion of AE to AF is determined by the difference of their Times onely Seeing therefore AB AE as also AC AF are contemporary the difference of the Times in which AB and AC are described is the same with that in which AE and AF are described Wherfore the proportion of AB to AC and the proportion of AE to AF are both determined by the same Cause But the Cause which so determines the proportion of both works equally in equal Times for it is Uniform Motion and therefore by the last precedent Definition the proportion of AB to AC is the same with that of AE to AF and consequently AB AC AF. AF are Proportionals which is the first Secondly taking the Antecedents in different Lines I say AB AE AC AF are Proportionals For seeing AB AE are described in the same Time the difference of the Velocities in which they are described are the sole Cause of the proportion they have to one another And the same may be said of the proportion of AC to AF. But seeing both the Lines AD and AG are passed over by Uniform Motion the difference of the Velocities in which AB AE are described will be the same with the
FHG and the straight Lines DA DB and DC proportional to the straight Lines HE HF and HG I say the three points A B and C have Like Situation with the three points E F G or are placed Alike For if HE be understood to be layed upon DA so that the point H be in D the point F will be in the straight Line DB by reason of the equality of the Angles ADB and EHF and the point G will be in the straight Line DC by reason of the equality of the Angles BDC and FHG and the strright Lines AB and EF as also BC and FG will be parallel because AD. ED BH FH CD GH are Proportionals by construction and therefore the distances between the points A and B and the points B and C will be proportional to the distances between the points E and F and the points F and G. Wherefore in the situation of the points A B and C and the situation of the points E F and G the Angles in the same order are equal so that their situations differ in nothing but the inequality of their distances from one another and of their distances from the points D and H. Now in both the orders of Points those inequalities are equal for AB BC EF. FG which are their distances from one another as also DA. DB. DC HE. HF. HG which are their distances from the assumed points D and H are Proportionals Their difference therefore consists solely in the magnitude of their distances But by the definition of Like Chap. 11. Art 2 those things which differ onely in Magnitude are Like Wherefore the points A B and C have to one another Like Situation with the points E F and G or are placed Alike which was to be proved FIGURE is quantity determined by the Situation or placing of all its extreme Points Now I call those points Extreme which are contiguous to the place which is without the figure In Lines therefore and Superficies all Points may be called Extreme but in Solids onely those which are in the Superficies that includes them Like Figures are those whose extreme points in one of them are all placed like all the extreme points in the other for such Figures differ in nothing but Magnitude And like Figures are alike placed when in both of them the homologal straight lines that is the straight lines which connect the points which answer one another are parallel and have their proportional sides enclined the same way And seeing every Straight Line is like every other Straight Line and every Plain like every other Plain when nothing but Plainness is considered if the Lines which include Plains or the Superficies which include Solids have their proportions known it will not be hard to know whether any Figure be like or unlike to another propounded Figure And thus much concerning the First Grounds of Philosophy The next place belongs to Geometry in which the Quantities of Figures are sought out from the Proportions of Lines and Angles Wherefore it is necessary for him that would study Geometry to know first what is the nature of Quantity Proportion Angle and Figure Having therefore explained these in the three last Chapters I thought â—it to add them to this Part and so passe to the next OF THE PROPORTIONS OF MOTIONS AND MAGNITVDES CHAP. XV. Of the Nature Properties and diverse Considerations of Motion and Endeavour 1 Repetition of some Principles of the doctrine of Motion formerly set down 2 Other Principles added to them 3 Certain Theoremes concerning the nature of Motion 4 Diverse Considerations of Motion 5 The way by which the first Endeavour of Bodies Moved â—endoth 6 In Motion which is made by Concourse one of the Movents ceasing the Endeavour is made by the way by which the rest tend 7 The Endeavour of any Moved Body which having its Motion in the Circumference of a Circle parts from the same proceeds afterwards in a straight line which toucheth the Circle 8 How much greater the Velocity or Magnitude is of a Movent so much greater is the Efficacy thereof upon any other Body in its way 1 THe next things in order to be treated of are MOTION and MAGNITUDE which are the most common Accidents of all Bodies This place therefore most properly belongs to the Elements of Geometry But because this part of Philosophy having been improved by the best Wits of all Ages has afforded greater plenty of matter then can well be thrust together within the narrow limits of this discourse I thought fit to admonish the Reader that before he proceed further he take into his hands the Works of Euclide Archimedes Apollonius and other as well Ancient as Modern Writers For to what end is it to do over again that which is already done The little therefore that I shall say concerning Geometry in some of the following Chapters shall be such onely as is new and conducing to Natural Philosophy I have already delivered some of the Principles of this doctrine in the 8 9 Chapters which I shall briefly put together here that the Reader in going on may have their light neerer at hand First therefore in the 8th Chap. and 10 Article Motion is defined to be the continual privation of one place and acquisition of another Secondly it is there shewn that Whatsoever is Moved is Moved in Time Thirdly in the same Chap. 11. Article I have defined Rest to be when a Body remains for some time in one place Fourthly it is there shewn that Whatsoever is Moved is not in any determined place as also that the same has been Moved is still Moved and will yet be Moved So that in every part of that Space in which Motion is made we may consider three Times namely the Past the Present and the Future Time Fiftly in the 15 Article of the same Chapter I have defined Velocity or Swiftness to be Motion considered as Power namely that Power by which a Body Moved may in a certain Time transmit a certain Length which also may more briefly be enunciated thus Velocity is the quantity of Motion determined by Time and Line Sixthly in the same Chap. 16. Article I have shewn that Motion is the Measure of Time Seventhly in the same Chap. 17th Art I have defined Motions to be Equally Swift when in Equal Times Equal Lengths are transmitted by them Eighthly in the 18 Article of the same Chapter Motions are defined to be Equal when the Swiftness of one Moved Body computed in every part of its magnitude is equal to the Swiftness of another computed also in every part of its magnitude From whence it is to be noted that Motions Equal to one another and Motions Equally Swift do not signifie the same thing for when two horses draw abrest the Motion of both is greater then the Motion of either of them singly but the Swiftness of both together is but Equal to that of either Ninthly in the 19 Article of the
same Chapter I have shewn that Whatsoever is at Rest will alwayes be at Rest unless there be some other Body besides it which by getting into its place suffers it no longer to remain at Rest. And that Whatsoever is Moved will alwayes be Moved unless there be some other Body besides it which hinders its Motion Tenthly In the 9 Chapter and 7 Article I have demonstrated that When any Body is moved which was formerly at Rest the immediate efficient cause of that Motion is in some other Moved and Contiguou◠Body Eleventhly I have shewn in the same place that Whatsoever is Moved will always be Moved in the same way and with the same Swiftness if it be not hindered by some other Moved and Contiguou◠Body 2 To which Principles I shall here add these that follow First I define ENDEAVOUR to be Motion made in less Space and Time then can be given that is less then can be determined or assigned by Exposition or Number that is Motion made through the length of a Point and in an Instant or Point of Time For the explayning of which Definition it must be remembred that by a Point is not to be understood that which has no quantity or which cannot by any means be divided for there is no such thing in Nature but that whose quantity is not at all considered that is whereof neither quantity nor any part is computed in demonstration so that a Point is not to be taken for an Indivisible but for an Undivided thing as also an Instant is to be taken for an Undivided and not for an Indivisible Time In like manner Endeavour is to be conceived as Motion but so as that neither the quantity of the Time in which nor of the Line in which it is made may in demonstration be at all brought into comparison with the quantity of that Time or of that Line of which it is a part And yet as a Point may be compared with a Point so one Endeavour may be compared with another Endeavour and one may be found to be greater or lesse then another For if the Vertical points of two Angles be compared they will be equal or unequal in the same proportion which the Angles themselves have to one another Or if a straight Line cut many Circumferences of Concentrick Circles the inequality of the points of intersection will be in the same proportion which the Perimeters have to one another And in the same manner if two Motions begin and end both together their Endeavours will be Equal or Unequal according to the proportion of their Velocities as we see a bullet of Lead descend with greater Endeavour then a ball of Wooll Secondly I define IMPETUS or Quickness of Motion to be the Swiftness or Velocity of the Body moved but considered in the several points of that time in which it is moved In which sense Impetus is nothing else but the quantity or velocity of Endeavour But considered with the whole time it is the whole velocity of the Body moved taken together throughout all the time and equal to the Product of a Line representing the time multiplyed into a Line representing the arithmetically mean Impetus or Quickness Which Arithmetical Mean what it is is defined in the 29th Article of the 13th Chapter And because in equal times the wayes that are passed are as the Velocities and the Impetus is the Velocity they go withal reckoned in all the several points of the times it followeth that during any time whatsoever howsoever the Impetus be encreased or decreased the length of the way passed over shall be encreased or decreased in the same proportion and the same Line shall represent both the way of the Body moved and the several Impetus or degrees of Swiftness wherewith the way is passed over And if the Body moved be not a point but a straight line moved so as that every point thereof make a several straight line the Plain described by its motion whether Uniform Accelerated or Retarded shall be greater or less the time being the same in the same proportion with that of the Impetus reckoned in one motion to the Impetus reckoned in the other For the reason is the same in Parallelograms and their Sides For the same cause also if the Body moved be a Plain the Solid described shall be still greater or less in the proportions of the several Impetus or Quicknesses reckoned through one Line to the several Impetus reckoned through another This understood let ABCD in the first figure of the 17th Chapter be a Parallelogram in which suppose the side AB to be moved parallelly to the opposite side CD decreasing al the way till it vanish in the point C and so describing the figure ABEFC the point B as AB decreaseth will therefore describe the Line BEFC and suppose the time of this motion designed by the line CD and in the same time CD suppose the side AC to be moved parallelly and uniformly to BD. From the point O taken at adventure in the Line CD draw OR parallel to BD cutting the Line BEFC in E and the side AB in R. And again from the point Q taken also at adventure in the Line CD draw QS parallel to BD cutting the Line BEFC in F and the side AB in S and draw EG and FH parallel to CD cutting AC in G and H. Lastly suppose the same construction done in all the points possible of the Line BEFC I sa◠that as the proportions of the Swiftnesses wherewith QF OE DB and all the rest supopsed to be drawn parallel to DB and terminated in the Line BEFC are to the proportions of their several Times designed by the several parallels HF GE AB and all the rest supposed to be drawn parallel to the Line of time CD and terminated in the Line BEFC the aggregate to the aggregate so is the Area or Plain DBEFC to the Area or Plain ACFEB For as AB decreasing continually by the line BEFC vanisheth in the time CD into the point C so in the same time the line DC continually decreasing vanisheth by the same line CFEB into the point B and the point D describeth in that decreasing motion the line DB equall to the line AC described by the point A in the decreasing motion of A B their swiftnesses are therefore equal Again because in the time GE the point O describeth the line OE and in the same time the point R describeth the line RE the line OE shall be to the line RE as the swiftness wherewith OE is described to the swiftness wherwith RE is described In like māner because in the same time HF the point Q describeth the Line QF and the point S the Line SF it shall be as the swiftness by which QF is described to the swiftness by which SF is described so the Line it self QF to the Line it self SF and so in all the Lines that can possibly be
point F of the Time A F the Impetus acquired be F K and let D E be the Length passed through in the Time A B with Impetus Uniformly encreased I say the Length D E is to the Length transmitted in the Time A F as the Time A B multiplyed into the Mean of the Impetus encreasing through the time A B is to the Time A F multiplyed into the Mean of the Impetus encreasing through the time A F. For seeing the Triangle A B I is the whole Velocity of the Body moved in the Time A B till the Impetus acquired be B I and the Triangle A F K the whole Velocity of the Body moved in the Time A F with Impetus encreasing till there be acquired the Impetus F K the Length D E to the Length acquired in the Time A F with Impetus encreasing from Rest in A till there be acquired the Impetus F K will be as the Triangle A B I to the Triangle A F K that is if the Triangles A B I and A F K be like in duplicate proportion of the Time A B to the Time A F but if unlike in the proportion compounded of the proportions of A B to B I of A K to A F. Wherefore as A B I is to A F K so let D E be to D P for so the Length transmitted in the Time A B with Impetus encreasing to B I will be to the Length transmitted in the Time A F with Impetus encreasing to F K as the triangle A B I is to the triangle A F K But the triangle A B I is made by the multiplication of the Time A B into the Mean of the Impetus encreasing to B I and the triangle A F K is made by the multiplication of the Time A F into the Mean of the Impetus encreasing to F K and therefore the Length D E which is transmitted in the Time A B with Impetus encreasing to B I to the Length D P which is transmitted in the Time A F with Impetus encreasing to F K is as the product which is made of the Time A B multiplyed into its mean Impetus to the product of the Time A F multiplyed also into its mean Impetus which was to be proved Corol. 1 In Motion Uniformly accelerated the proportion of the Lengths transmitted to that of their Times is compounded of the proportions of their Times to their Times and Impetus to Impetus Corol. 2 In Motion Uniformly accelerated the Lengths transmitted in equal times taken in continual succession from the beginning of Motion are as the differences of square numbers beginning from Unity namely as 3 5 7 c. For if in the first time the Length transmitted be as 1 in the first and second times the Length transmitted will be as 4 which is the Square of 2 and in the three first times it will be as 9 which is the Square of 3 and in the four first times as 16 and so on Now the differences of these Squares are 3 5 7 c. Corol. 3 In Motion Uniformly accelerated from Rest the Length transmitted is to another Length transmitted vniformly in the same Time but with such Impetus as was acquired by the accelerated Motion in the last point of that Time as a triangle to a parallelogram which have their altitude and base common For seeing the Length D E in the same 1 figure is passed through with Velocity as the triangle A B I it is necessary that for the passing through of a Length which is double to D E the Velocity be as the parallelogram A I for the parallelogram A I is double to the triangle A B I. 4 In Motion which beginning from Rest is so accelerated that the Impetus thereof encrease continually in proportion duplicate to the proportion of the times in which it is made a Length transmitted in one time will be to a Length transmitted in another time as the product made by the Mean Impetus multiplyed into the time of one of those Motions to the product of the Mean Impetus multiplyed into the time of the other Motion For let A B in the 2d figure represent a Time in whose first instant A let the Impetus be as the point A but as the time proceeds so let the Impetus encrease continually in duplicate proportion to that of the times till in the last point of time B the Impetus acquired be B I then taking the point F any where in the time A B let the Impetus F K acquired in the time A F be ordinately applyed to that point F. Seeing therefore the proportion of F K to B I is supposed to be duplicate to that of A F to A B the proportion of A F to A B will be subduplicate to that of F K to B I and that of A B to A F will be by Chap. 13. Article 16 duplicate to that of B I to F K and consequently the point K will be in a parabolical line whose diameter is A B and base B I and for the same reason to what point soever of the time A B the Impetus acquired in that time be ordinately applyed the straight line designing that Impetus will be in the same parabolical line A K I. Wherefore the mean Impetus multiplyed into the whole time A B will be the Parabola A K I B equal to the parallelogram A M which parallelogram has for one side the line of time A B and for the other the line of the Impetus A L which is two thirds of the Impetus B I for every Parabola is equal to two thirds of that parallelogram with which it has its altitude and base common Wherefore the whole Velocity in A B will be the parallelogram A M as being made by the multiplication of the Impetus A L into the time A B. And in like manner if F N be taken which is two thirds of the Impetus F K and the parallelogram F O be completed F O will be the whole Velocity in the time A F as being made by the Uniform Impetus A O or F N multiplyed into the time A F. Let now the length transmitted in the time A B and with the Velocity A M be the straight line D E and lastly let the Length transmitted in the time A F with the Velocity A N be D P I say that as A M is to A N or as the Parabola A K I B to the Parabola A F K so is D E to D P. For as A M is to F L that is as A B is to A F so let D E be to D G. Now the proportion of A M to A N is compounded of the proportions of A M to F L and of F L to A N. But as A M to F L so by construction is D E to D G and as F L is to A N seeing the time in both is the same namely A F so is the
been shewn at the end of the 3d Article of the 15th Chapter The Cause therefore of their Restitution is some motion either of the parts of the Ambient or of the parts of the Body compressed or extended But the parts of the Ambient have no endeavour which contributes to their Compression or Extension nor to the setting of them at liberty or Restitution It remayns therefore that from the time of their Compression or Extension there be left some endeavour or motion by which the impediment being removed every part resumes its former place that is to say the whole Restores it self 14 In the Carriage of Bodies if that Body which carries another hit upon any obstacle or be by any means suddenly stopped and that which is carried be not stopped it will go on till its motion be by some external impediment taken away For I have demonstrated in the 8th Chapter at the 19th Article that Motion unless it be hindred by some external resistance will be continued eternally with the same celerity and in the 7th Article of the 9th Chap. that the action of an external Agent is of no effect without contact When therefore that which carrieth another thing is stopped that stop doth not presently take away the motion of that which is carried It will therefore proceed till its motion be by little and little extinguished by some external resistance Which was to be proved Though experience alone had been sufficient to prove this In like manner if that Body which carrieth another be put from rest into sudden motion that which is carried will not be moved forwards together with it but will be left behind For the contiguous part of the Body carried hath almost the same motion with the Body which carries it and the remote parts will receive different Velocities according to their different distances from the Body that carries them namely the more remote the parts are the less will be their degrees of Velocity It is necessary therefore that the Body which is carried be left accordingly more or less behind And this also is manifest by experience when at the starting forward of the Horse the Rider falleth backwards 15 In Percussion therefore when one hard Body is in some small â—art of it stricken by another with great force it is not necessary that the whole Body should yeild to the stroke with the same celerity with which the stricken part yeilds For the rest of the parts receive their motion from the motion of the part stricken and yeilding which motion is less propagated every way towards the sides then it is directly forwards And hence it is that sometimes very hard Bodies which being erected can hardly be made to stand are more easily broken then thrown down by a violent stroke when nevertheless if all their parts together were by any weak motion thrust forwards they would easily be cast down 16 Though the difference between Trusion and Percussion consist onely in this that in Trusion the motion both of the Movent and Moved Body begin both together in their very contact and in Percussion the striking Body is first moved and afterwards the Body stricken Yet their Effects are so different that it seems scarce possible to compare their forces with one another I say any effect of Percussion being propounded as for example the stroke of a Beetle of any weight assigned by which a Pile of any given length is to be driven into earth of any tenacity given it seems to me very hard if not impossible to define with what weight or with what stroke and in what time the same pile may be driven 〈◊〉 a depth assigned into the same earth The cause of which difficulty is this that the velocity of the Percutient is to be compared with the magnitude of the Ponderant Now Velocity seeing it is computed by the length of space transmitted is to be accounted but as one Dimension but Waight is as a solid thing being measured by the dimension of the whole Body And there is no comparison to be made of a Solid Body with a Length that is with a Line 17 If the internal parts of a Body be at rest or retain the same situation with one another for any time how little soever there cannot in those parts be generated any new motion or endeavour whereof the efficient cause is not without the Body of which they are parts For if any small part which is comprehended within the Superficies of the whole Body be supposed to be now at rest and by and by to be moved that part must of necessity receive its motion from some moved and contiguous Body But by supposition there is no such moved and contiguous part within the Body Wherefore if there be any Endeavour or Motion or change of situation in the internal parts of that Body it must needs arise from some efficient cause that is without the Body which contains them Which was to be proved 18 In hard Bodies therefore which are compressed or extended if that which compresseth or extendeth them being taken away they restore themselves to their former place or situation it must needs be that that Endeavour or Motion of their internal parts by which they were able to recover their former places or situations was not extinguished when the force by which they were compressed or extended was taken away Therefore when the Lath of a Cross-bow bent doth as soon as it is at liberty restore it self though to him that judges by Sense both it and all its parts seem to be at rest yet he that judging by Reason doth not account the taking away of impediment for an efficient cause nor conceives that without an efficient cause any thing can pass from Rest to Motion will conclude that the parts were already in motion before they began to restore themselves 19 Action and Reaction proceed in the same Line but from opposite Terms For seeing Reaction is nothing but Endeavour in the Patient to restore it self to that situation from which it was forced by the Agent the endeavour or motion both of the Agent and Patient or Reagent will be propagated between the same terms yet so as that in Action the Term from which is in Reaction the Term to which And seeing all Action proceeds in this manner not onely between the opposite Terms of the whole line in which it is propagated but also in all the parts of that line the Terms from which and to which both of the Action and Reaction will be in the same line Wherefore Action and Reaction proceed in the same line c. 20 To what has been said of Motion I will add what I have to say concerning Habit. Habit therefore is a generation of Motion not of Motion simply but an easie conducting of the moved Body in a certain and designed way And seeing it is attained by the weakning of such endeavours as divert its motion therefore such endeavours are to be
that which the Line of Incidence makes with that Line which from the point of Refraction is drawn perpendicular to the separating Superficies 8 The Angle of Incidence is the Complement to a right Angle of the Angle of Inclination And so in the first Figure the Refraction is made in A B F. The Refracted Line is B F. The Line of Incidence is A B. The Point of Incidence and of Refraction is B. The Refracting or Separating Superficies is D B E. The Line of Incidence produced directly is A B C The Perpendicular to the separating Superficies is B H. The Angle of Refraction is C B F. The Angle Refracted is H B F. The Angle of Inclination is A B G or H B C. The Angle of Incidence is A B D. 9 Moreover the Thinner Medium is understood to be that in which there is less resistance to Motion or to the generation of Motion the Thicker that wherin there is greater resistance 10 And that Medium in which there is equal resistance every where is a Homogeneous Medium All other Mediums are Heterogeneous 2 If a Body pass or there be generation of Motion from one Medium to another of different Density in a line perpendicular to the Separating Superficies there will be no Refraction For seeing on every side of the perdendicular all things in the Mediums are supposed to be like and equal if the Motion it self be supposed to be perpendicular the Inclinations also will be equal or rather none at all and therefore there can be no cause from which Refraction may be inferred to be on one side of the perpendicular which wil not cōclude the same Refraction to be on the other side Which being so Refraction on one side will destroy Refraction on the other side and consequently either the Refracted line will be every where which is absurd or there will be no Refracted line at all which was to be demonstrated Corol. It is manifest from hence that the cause of Refraction consisteth onely in the obliquity of the line of Incidence whether the Incident Body penetrate both the Mediums or without penetrating propagate motion by Pressure onely 3 If a Body without any change of situation of its internal parts as a stone be moved obliquely out of the thinner Medium and proceed penetrating the thicker Medium and the thicker Medium be such as that its internal parts being moved restore themselves to their former situation the angle Refracted will be greater then the angle of Inclination For let D B E in the same first figure be the separating Superficies of two Mediums and let a Body as a stone thrown be understood to be moved as is supposed in the straight line A B C and let A B be in the thinner Medium as in the Aire and B C in the thicker as in the Water I say the stone w ch being thrown is moved in the line A B will not proceed in the line B C but in some other line namely that with which the perpendicular B H makes the Refracted angle H B F greater then the angle of Inclination H B C. For seeing the stone coming from A and falling upon B makes that which is at B proceed towards H and that the like is done in all the straight lines which are parallel to B H and seeing the parts moved restore themselves by contrary motion in the same line there will be contrary motion generated in H B and in all the straight lines which are parallel to it Wherefore the motion of the stone will be made by the concourse of the motions in A G that is in D B and in G B that is in B H and lastly in H B that is by the concourse of three motions But by the concourse of the motions in A G and B H the stone will be carried to C and therefore by adding the motion in H B it will be carried higher in some other line as in B F and make the angle H B F greater then the angle H B C. And from hence may be derived the cause why Bodies which are thrown in a very oblique line if either they be any thing flat or be thrown with great force will when they fall upon the water be cast up again from the water into the aire For let A B in the 2d figure be the superficies of the water into which from the point C let a stone be thrown in the straight line C A making with the line B A produced a very little angle C A D and producing B A indefinitely to D let C D be drawn perpendicular to it and A E parallel to C D. The stone therefore will be moved in C A by the concourse of two motions in C D and D A whose velocities are as the lines themselves C D and D A. And from the motion in C D and all its parallels downwards as soon as the stone falls upon A there will be Reaction upwards because the water restores it self to its former situation If now the stone be thrown with sufficient obliquity that is if the straight line C D be short enough that is if the endeavour of the stone downwards be less then the Reaction of the water upwards that is less then the endeavour it hath from its own gravity for that may be the stone will by reason of the excess of the endeavour which the water hath to restore it self above that which the stone hath downwards be raised again above the Superficies A B and be carried higher being reflected in a line which goes higher as the line A G. 4 If from a point whatsoever the Medium be Endeavour be propagated every way into all the parts of that Medium and to the same Endeavour there be obliquely opposed another Medium of a different nature that is either thinner or thicker that Endeavour will be so refracted that the sine of the angle Refracted to the sine of the angle of Inclination will be as the density of the first Medium to the density of the second Medium reciprocally taken First let a Body be in the thinner Medium in A Figure 3d. and let it be understood to have endeavour every way and consequently that its endeavour proceed in the lines A B and A b to which let B b the superficies of the thicker Medium be obliquely opposed in B and b so that A B and A b be equal and let the straight line B b be produced both wayes From the points B and b let the perpendiculars B C and b c be drawn and upon the centers B and b and at the equal distances B A and b A let the Circles A C and A c be described cutting B C and b c in C and c and the same C B and c b produced in D and d as also A B and A b produced in E and e. Then from the point A to the straight lines B C and b c let the perpendiculars
A F and A f be drawn A F therefore will be the sine of the angle of Inclination of the straight line A B and A f the sine of the angle of Inclination of the straight line A h which two Inclinations are by construction made equal I say as the density of the Medium in which are B C and b c is to the density of the Medium in which are B D and b d so is the sine of the angle Refracted to the sine of the angle of Inclination Let the straight line F G be drawn parallel to the straight line A B meeting with the straight line b B produced in G. Seeing therefore A F and B G are also parallels they will be equal and consequently the endeavour in A F is propagated in the same time in which the endeavour in B G would be propagated if the Medium were of the same density But because B G is in a thicker Medium that is in a Medium which resists the endeavour more then the Medium in which A F is the endeavour will be propagated less in B G then in A F according to the proportion which the density of the Medium in which A F is hath to the density of the Medium in which B G is Let therefore the density of the Medium in which B G is be to the density of the Medium in which A F is as B G is to B H and let the measure of the time be the Radius of the Circle Let H I be drawn parallel to B D meeting with the circumference in I and from the point I let I K be drawn perpendicular to B D which being done B H and I K will be equal and I K will be to A F as the density of the Medium in which is A F is to the density of the Medium in which is I K. Seeing therefore in the time A B which is the Radius of the Circle the endeavour is propagated in A F in the thinner Medium it will be propagated in the same time that is in the time B I in the thicker Medium from K to I. Therefore B I is the Refracted line of the line of Incidence A B and I K is the sine of the angle Refracted and A F the sine of the angle of Inclination Wherefore seeing I K is to A F as the density of the Medium in which is A F to the density of the Medium in which is I K it will be as the density of the Medium in which is A F or B C to the density of the Medium in which is I K or B D so the sine of the angle Refracted to the sine of the angle of Inclination And by the same reason it may be shewn that as the density of the thinner Medium is to the density of the thicker Medium so will K I the sine of the angle Refracted be to A F the sine of the Angle of Inclination Secondly let the Body which endeavoureth every way be in the thicker Medium at I. If therefore both the Mediums were of the same density the endeavour of the Body in I B would tend directly to L and the sine of the angle of Inclination L M would be equal to I K or B H. But because the density of the Medium in which is IK to the density of the Medium in which is L M is as BH to B G that is to A F the endeavour will be propagated further in the Medium in which L M is then in the Medium in which I K is in the proportion of density to density that is of M L to A F. Wherefore B A being drawn the angle Refracted will be C B A and its sine A F. But L M is the sine of the angle of Inclination and therefore again as the density of one Medium is to the density of the different Medium so reciprocally is the sine of the angle Refracted to the sine of the angle of Inclination which was to be demonstrated In this Demonstration I have made the separating Superficies B b plain by construction But though it were concave or convex the Theoreme would nevertheless be true For the Refraction being made in the point B of the plain separating Superficies if a crooked line as P Q be drawn touching the separating line in the point B neither the Refracted line B I nor the perpendicular B D will be altered and the Refracted angle K B I as also its sine K I will be still the same they were 5 The sine of the angle Refracted in one Inclination is to the sine of the angle Refracted in another Inclination as the sine of the angle of that Inclination to the sine of the angle of this Inclination For seeing the sine of the Refracted angle is to the sine of the angle of Inclination whatsoever that Inclination be as the density of one Medium to the density of the other Medium the proportion of the sine of the Refracted angle to the sine of the angle of Inclination will be compounded of the proportions of density to density and of the sine of the angle of one Inclination to the sine of the angle of the other Inclination But the proportions of the densities in the same Homogeneous Body are supposed to be the same Wherefore Refracted angles in different Inclinations are as the sines of the angles of those Inclinations which was to be demonstrated 6 If two lines of Incidence having equal inclination be the one in a thinner the other in a thicker Medium the sine of the angle of their Inclination will be a mean proportional between the two sines of their angles Refracted For let the straight line AB in the same 3d figure have its Inclination in the thinner Medium and be refracted in the thicker Medium in B I and let E B have as much Inclination in the thicker Medium and be refracted in the thinner Medium in B S and let R S the sine of the angle Refracted be drawn I say the straight lines R S A F and I K are in continual proportion For it is as the density of the thicker Medium to the density of the thinner Medium so R S to A F. But it is also as the density of the same thicker Medium to that of the same thinner Medium so AF to IK Wherefore R S. A F A F. I K are propoortionals that is R S A F and I K are in continual proportion and A F is the Mean proportional which was to be proved 7 If the angle of Inclination be semirect and the line of Inclination be in the thicker Medium and the proportion of the Densities be as that of a Diagonal to the side of its Square and the separating Superficies be plain the Refracted line will be in that separating Superficies For in the Circle A C in the 4th figure let the angle
will remain no cause at all why the water should be forced out Wherefore the assertion of Vacuum is repugnant to the very experiment which is here brought to establish it Many other Phaenomena are usually brought for Vacuum as those of Weather-glasses Aeolipiles Wind-guns c. Which would all be very hard to be salved unless water be penetrable by aire without the intermixture of empty space But now seeing aire may with no great endeavour pass through not onely water but any other fluid Body though never so stubborn as Quicksilver these Phaenomena prove nothing Nevertheless it might in reason be expected that he that would take away Vacuum should without Vacuum shew us such causes of these Phaenomena as should be at least of equal if not greater probability This therefore shall be done in the following discourse when I come to speak of these Phaenomena in their proper places But first the most general Hypotheses of natural Philosophy are to be premised And seeing that Suppositions are put for the true Causes of apparent Effects every Supposition except such as be absurd must of necessity consist of some supposed possible Motion for Rest can never be the Essicient Cause of any thing Motion supposeth Bodies Moveable of which there are three kinds Fluid Consistent and mixt of both Fluid are those whose parts may by very weak endeavonr be separated from one another and Consistent those for the separation of whose parts greater force is to be applyed There are therefore degrees of Consistency which degrees by comparison with more or less Consistent have the names of Hardness or Softness Wherefore a Fluid Body is alwayes divisible into Bodies equally Fluid as Quantity into Quantities and Soft Bodies of whatsoever degree of Softness into Soft Bodies of the same degree And though many men seem to conceive no other difference of Fluidity but such as ariseth from the different magnitudes of the parts in which Sense Dust though of Diamonds may be called Fluid Yet I understand by Fluidity that which is made such by Nature equally in every part of the Fluid Body not as Dust is Fluid for so a House which is falling in pieces may be called Fluid but in such manner as Water seems Fluid and to divide it self into parts perpetually Fluid And this being well understood I come to my Suppositions 5 First therefore I suppose That the Immense Space which we call the World is the Aggregate of all Bodies which are either Consistent Visible as the Earth and the Starres or Invisible as the small Atomes which are disseminated through the whole space between the Earth and the Stars and lastly that most Fluid Aether which so fils all the rest of the Universe as that it leaves in it no empty place at all Secondly I suppose with Copernicus That the greater Bodies of the World which are both consistent and permanent have such order amongst themselves as that the Sunne hath the first place Mercury the second Venus the third The Earth with the Moon going about it the fourth Mars the fifth Jupiter with his Attendants the sixth Saturne the seventh and after these the Fixed Starres have their several distances from the Sunne Thirdly I suppose That in the Sunne the rest of the Planets there is and alwayes has been a Simple Circular Motion Fourthly I suppose That in the Body of the Aire there are certain other Bodies intermingled which are not Fluid but withal that they are so small that they are not preceptible by Sense and that these also have their proper Simple Motion and are some of them more some less hard or consistent Fifthly I suppose with Kepler That as the distance between the Sunne and the Earth is to the distance between the Moon and the Earth so the distance between the Moon and the Earth is to the Semidiameter of the Earth As for the Magnitude of the Circles and the Times in which they are described by the Bodies which are in them I will suppose them to be such as shall seem most agreeable to the Phaenomena in question 6 The causes of the different Seasons of the Year and of the several variations of Dayes and Nights in all the parts of the superficies of the Earth have been demonstrated first by Copernicus and since by Kepler Galilaeus and others from the supposition of the Earths diurnal revolution about its own Axis together with its Annual motion about the Sunne in the Ecliptick according to the order of the Signes and thirdly by the annual revolution of the same Earth about its own center contrary to the order of the Signs I suppose with Copernicus That the diurnal revolution is from the motion of the Earth by which the Aequinoctial Circle is described about it And as for the other two annual motions they are the efficient cause of the Earths being carried about in the Ecliptick in such manner as that its Axis is alwayes kept parallel to it self Which parallelisme was for this reason introduced lest by the Earths annual revolution its Poles should seem to be necessarily carried about the Sunne contrary to experience I have in the 10th Artic. of the â—â—th Chap. demonstrated from the supposition of Simple Circular Motion in the Sun that the Earth is so carried about the Sunne as that its Axis is thereby kept always parallel to it self Wherefore from these two supposed motions in the Sunne the one Simple Circular Motion the other Circular Motion about its owns Center it may be demonstrated that the Year hath both the same variations of Dayes and Nights as have been demonstrated by Copernicus For if the Circle abcd in the 3d Figure be the Ecliptick whose Center is e and Diameter aec and the Earth be placed in a the Sunne be moved in the little Circle fghi namely according to the order f g h i it hath been demonstrated that a Body placed in a will be moved in the same order through the points of the Ecliptick a b c d and will alwayes keep its Axis parallel to its self But if as I have supposed the Earth also be moved with Simple Circular Motion in a plain that passeth through a cutting the plain of the Ecliptick so as that the common section of both the plains be in ac thus also the Axis of the Earth will be kept alwayes parallel to it self For let the Center of the Earth be moved about in the Circumference of the Epicycle whose Diameter is lak which is a part of the straight line lac Therefore lak the Diameter of the Epicycle passing through the Center of the Earth will be in the plain of the Ecliptick Wherefore seeing that by reason of the Earths Simple Motion both in the Ecliptick and in its Epicycle the straight line lak is kept alwayes parallel to it self every other straight line also taken in the Body of the Earth and consequently its Axis will in like manner be kept alwayes parallel
reach the Zodiack of the fixed Starres wil fall stil upon the same fixed Starres because the whole Orbe a b c d is supposed to have no magnitude at all in respect of the great distance of the fixed Starres Supposing now the Sun to be in c it remains that I shew the cause why the Earth is neerer to the Sunne when in its annual motion it is found to be in d then when it is in b. And I take the cause to be this When the Earth is in the beginning of Capricorn at b the Sunne appears in the beginning of Cancer at d then is the midst of Summer But in the midst of Summer the Northern parts of the Earth are towards the Sunne which is almost all dry land containing all Europe and much the greatest part of Asia and America But when the Earth is in the beginning of Cancer at d it is the midst of Winter and that part of the Earth is towards the Sunne which contains those great Seas called the South Sea and the Indian Sea which are of farre greater extent then all the dry Land in that Hemisphere Wherefore by the last Article of the 21 Chapter when the Earth is in d it will come neerer to its first Movent that is to the Sunne which is in t that is to say the Earth is neerer to the Sunne in the midst of Winter when it is in d then in the midst of Summer when it is b and therefore during the Winter the Sunne is in its Perigaeum and in its Apogaeum during the Summer And thus I have shewn a possible cause of the Excentricity of the Earth which was to be done I am therefore of Keplers opinion in this that he attributes the Excentricity of the Earth to the difference of the parts thereof and supposes one part to be affected and another disaffected to the Sunne And I dissent from him in this that he thinks it to be by Magnetick virtue and that this Magnetick virtue or attraction and thrusting back of the Earth is wrought by immateriate Species which cannot be because nothing can give motion but a Body moved and contiguous For if those Bodies be not moved which are contiguous to a Body unmoved how this Body should begin to be moved is not imaginable as has been demonstrated in the 7th Article of the 9th Chapter and often inculcated in other places to the end that Philosophers might at last abstain from the use of such unconceiveable connexions of words I dissent also from him in this that he says the similitude of Bodies is the cause of their mutual attraction For if it were so I see no reason why one Egg should not be attracted by another If therefore one part of the Earth be more affected by the Sunne then another part it proceeds from this that one part hath more water the other more dry land And from hence it is as I shewed above that the Earth comes neerer to the Sunne when it shines upon that part where there is more water then when it shines upon that where there is more dry Land 9 This Excentricity of the Earth is the cause why the way of its annual motion is not a perfect Circle but either an Elliptical or almost an Elliptical line as also why the Axis of the Earth is not kept exactly parallel to it self in all places but onely in the Equinoctial points Now seeing I have said that the Moon is carried about by the Earth in the same manner that the Earth is by the Sunne and that the Earth goeth about the Sunne in such manner as that it shews sometimes one Hemisphere sometimes the other to the Sunne it remains to be enquired why the Moon has alwayes one and the same face turned towards the Earth Suppose therefore the Sunne to be moved with Simple Motion in the little Circle f g h i in the fourth figure whose Center is t and let ♈ ♋ ♎ ♑ be the annuall Circle of the Earth and a the beginning of Libra About the point a let the little Circle l k be described and in it let the Center of the Earth be understood to be moved with Simple motion and both the Sunne the Earth to be moved according to the order of the Signes Upon the Center a let the way of the Moon m n o p be described and let q r be the Diameter of a Circle cutting the Globe of the Moon into two Hemispheres whereof one is seen by us when the Moon is at the full and the other is turned from us The Diameter therefore of the Moon q o r will be perpendicular to the Straight Line t a. Wherefore the Moon is carried by reason of the Motion of the Earth from o towards p. But by reason of the motion of the Sunne if it were in p it would at the same time be carried from p towards o and by these two contrary Movents the straight line q r will be turned about and in a Quadrant of the Circle m n o p it will be turned so much as makes the fourth part of its whole conversion Wherefore when the Moon is in p q r will be parallel to the straight line m o. Secondly when the Moon is in m the straight line q r will by reason of the motion of the Earth be in m o. But by the working of the Suns motion upon it in the quadrant p m to◠same q r will be turned so much as makes another quarter of its whole conversion When therefore the Moon is in m q r will be perpendicular to the straight line o m. By the same reason when the Moon is in n q r will be parallel to the straight line m o and the Moon returning to o the same q r will return to its first place and the Body of the Moon will in one entire period make also one entire conversion upon her own Axis In the making of which it is manifest that one and the same face of the Moon is always turned towards the Earth And if any Diameter were taken in that little Circle in which the Moon were supposed to be carried about with simple motion the same effect would follow for if there were no action from the Sun every Diameter of the Moon would be carried about always parallel to it self Wherefore I have given a possible cause why one and the same face of the Moon is alwayes turned towards the Earth But it is to be noted that when the Moon is without the Ecliptick we do not alwayes see the same face precisely For we see onely that part which is illuminated But when the Moon is without the Ecliptick that part which is towards us is not exactly the same with that which is illuminated 10 To these three simple motions one of the Sunne another of the Moon and the third of the Earth in their own little Circles f g h i l k q r together with the Diurnal
of this Evocation and Swelling and such as agreeth with the rest of the Phaenomena of Heat may be thought to have given the cause of the Heat of the Sunne It hath been shewn in the 5 article of the 21 chapter that the fluid Medium which we call the Aire is so moved by the simple circular motion of the Sunne as that all its parts even the least do perpetually change places with one another which change of places is that which there I called Fermentation From this Fermentation of the Aire I have in the 8 article of the last chapter demonstrated that the water may be drawn up into the clouds And I shall now shew that the fluid parts may in like manner by the same Fermentation be drawn out from the internall to the externall parts of our Bodies For seeing that wheresoever the fluid Medium is contiguous to the Body of any living creature there the parts of that Medium are by perpetuall change of place separated from one another the contiguous parts of the living creature must of necessity endeavour to enter into the spaces of the separated parts For otherwise those parts supposing there is no Vacuum would have no place to go into And therefore that which is most fluid and separable in the parts of the living creature which are contiguous to the Medium will go first out and into the place thereof will succeed such other parts as can most easily transpire through the poâ—es of the skin And from hence it is necessary that the rest of the parts which are not separated must all together be moved outwards for the keeping of all places full But this motion outwards of all parts together must of necessity press those parts of the ambient Aire which are ready to leave their places and therefore all the parts of the Body endeavouring at once that way makes the Body swell Wherefore a possible cause is given of Heat from the Sunne which was to be done 4 We have now seen how Light and Heat are generated Heat by the simple motion of the Medium making the parts perpetually change places with one another and Light by this that by the same simple motion Action is propagated in a straight line But when a Body hath its parts so moved that it sensibly both Heats and Shines at the same time then it is that we say Fire is generated Now by Fire I do not understand a Body distinct from matter combustible or glowing as Wood or Iron but the matter it self not simply and always but then onely when it shineth and heateth He therefore that renders a cause possible and agreeable to the rest of the Phaenomena namely whence and from what action both the Shining and Heating proceed may be thought to have given a possible cause of the generation of Fire Let therefore ABC in the first Figure be a Sphere or the portion of a Sphere whose Center is D and let it be transparent and homogeneous as Cristal Glass or Water and objected to the Sunne Wherefore the foremost part ABC will by the simple motion of the Sunne by which it thrusts forwards the Medium be wrought upon by the Sun-beams in the straight lines EA FB and GC which straight lines may in respect of the great distance of the Sunne be taken for parallels And seeing the Medium within the Sphere is thicker then the Medium without it those Beams will be refracted towards their perpendiculars Let the straight lines EA and GC be produced till they cut the Sphere in H and I and drawing the perpendiculars AD and CD the refracted Beams EA and GC will of necessity fall the one between AH and AD the other between CI and CD Let those refracted Beams be AK and CL. And again let the lines DKM DLN be drawn perâ—endicular to the Sphere and let AK and CL be produced till they meet with the straight line BD produced in O. Seeing therefore the Medium within the Sphere is thicker then that without it the refracted line AK will recede further from the perpendicular KM then KO will recede from the same Wherefore KO will fall between the refracted line and the perpendicular Let therefore the refracted line be KP cutting FO in P and for the same reason the straight line LP will be the refracted line of the straight line CL. Wherfore seeing the Beams are nothing else but the Wayes in which the motion is propagated the motion about P will be so much more vehement then the motion about ABC by how much the base of the portion ABC is greater then the base of a like portion in the Sphere whose Center is P and whose magnitude is equal to that of the little Circle about P which comprehendeth all the Beams that are propagated from ABC and this Sphere being much less then the Sphere ABC the parts of the Medium that is of the Aire about P will change places with one another with much greater celerity then those about ABC If therefore any matter Combustible that is to say such as may be easily dissipated be placed in P the parts of that matter if the proportion be great enough between AC and a like portion of the little circle about P wil be freed from their mutual cohaesion and being separated will acquire simple motion But vehement simple motion generates in the beholder a Phantasm of Lucid and Hot as I have before deâ—onstrated of the simple motion of the Sunne and therefore the combustible matter which is placed in P will be made Lucid and Hot that is to say will be Fire Wherefore I have rendered a possible cause of Fire which was to be done 5 From the manner by which the Sunne generateth Fire it is easy to explaine the manner by which Fire may be generated by the collision of two Flints For by that Collision some of those particles of which the stone is compacted are violently separated and thrown off and being withall swiftly turned round the Eie is moved by them as it is in the generation of Light by the Sunne Wherefore they shine and falling upon matter which is already halfe dissipated such as is Tinder they throughly dissipate the parts thereof and make them turn round From whence as I have newly shewn Light and Heat that is to say Fire is generated 6 The shining of Glow-worms some kinds of Rotten Wood and of a kinde of stone made at Bolognia may have one common cause namely the exposing of them to the hot Sunne We finde by experience that the Bolonian stone shines not unless it be so exposed and after it has been exposed it shines but for a little time namely as long as it retains a certain degree of heat And the cause may be that the parts of which it is made may together with heat have Simple Motion imprinted in them by the Sunne Which if it be so it is necessary that it shine in the dark as
of the Niter is generated that vehement Motion and Inflammation And lastly when there is no more action from the Niter that is to say when the volatile parts of the Niter are flown out there is found about the sides a certain white substance which being thrown again into the fire will grow red hot again but will not be dissipated at least unlesse the fire be augmented If now a possible cause of this be found out the same will also be a possible cause why a grain of Gunpowder set on fire doth expand it selfe with such vehement motion and Shine And it may be caused in this manner Let the particles of which Niter consisteth be supposed to be some of them hard others watery and the rest aethereall Also let the hard particles be supposed to be spherically hollow like small bubbles so that many of them growing together may constitute a Body whose little cavernes are filled with a substance which is either watery or aethereal or both As soon therefore as the hard particles are dissipated the watery and aethereal particles will necessarily fly out and as they fly of necessity blow strongly the burning Coles and Brimstone which are mingled together whereupon there will follow a great expansion of Light with vehement flame and a violent dissipation of the particles of the Niter the Brimstone and the Coles Wherefore I have given a possible cause of the force of fired Gunpowder It is manifest from hence that for the rendering of the cause why a bullet of lead or iron shot from a peece of Ordnance flies with so great velocity there is no necessity to introduce such Rarefaction as by the common definition of it makes the same Matter to have sometimes more sometimes lesse Quantity which is unconceiveable For every thing is said to be greater or lesse as it hath more or lesse Quantity The violence with which a bullet is thrust out of a Gun proceeds from the swiftnesse of the small particles of the fired Powder at least it may proceed from that cause without the supposition of any Empty Space 11 Besides by the attrition or rubbing of one Body against another as of Wood against Wood we find that not only a certaine degree of Heat but Fire it selfe is sometimes generated For such motion is the reciprocation of pressure sometimes one way sometimes the other and by this reciprocation whatsoever is fluid in both the peeces of Wood is forced hither and thither and consequently to an endeavour of getting out and at last by breaking out makes Fire 12 Now Light is distinguished into First Second Third and so on infinitely And we call that First Light which is in the first Lucid Bodie as the Sunne Fire c. Second that which is in such Bodies as being not transparent are illuminated by the Sunne as the Moon a Wall c. and Third that which is in Bodies not transparent but illuminated by Second Light c. 13 Colour is Light but troubled Light namely such as is generated by perturbed motion as shall be made manifest by the Red Yellow Blew and Purple which are generated by the interposition of a Diaphanous Prisma whose opposite bases are triangular between the Light and that which is enlightened For let there be a Prisma of Glasse or of any other transparent matter which is of greater density then Aire and let the triangle ABC be the base of this Prisma Also let the straight line DE be the diameter of the Sunnes Body having oblique position to the straight line AB and let the Sunne-beames passe in the lines DA and EBC And lastly let the straight lines DA and EC be produced indefinitely to F and G. Seeing therefore the straight line DA by reason of the density of the Glasse is refracted towards the perpendicular let the line refracted at the point A be the straight line AH And againe seeing the Medium below AC is thinner then that above it the other refraction which will be made there will diverge from the perpendicular Let therefore this second refracted line be AI. Also let the same be done at the point C by making the first refracted line to be CK and the second CL. Seeing therefore the cause of the refraction in the point A of the straight line of AB is the excess of the resistance of the Medium in AB above the resistance of the Aire there must of necessity be reaction from the point A towards the point B and consequently the Medium at A within the triangle ABC will have its motion troubled that is to say the straight motion in AF and AH will be mixed with the transverse motion between the same AF and AH represented by the short transverse lines in the triangle AFH Againe seeing at the point A of the straight line AC there is a second refraction from AH in AI the motion of the Medium will againe be perturbed by reason of the transverse reaction from A towards C represented likewise by the short transverse lines in the triangle AHI And in the same manner there is a double perturbation represented by the transverse lines in the triangles CGK and CKL But as for the light between AI and CG it will not be perturbed because if there were in all the points of the straight lines AB and AC the same action which is in the points A and C then the plaine of the triangle CGK would be every where coincident with the plaine of the triangle AFH by which meanes all would appear alike between A and C. Besides it is to be observed that all the reaction at A tends towards the illuminated parts which are between A and C and consequently perturbeth the First Light And on the contrary that all the reaction at C tends towards the parts without the triangle or without the Prisma ABC where there is none but Second Light and that the triangle AFH shewes that perturbation of Light which is made in the Glasse it selfe as the triangle AHI shewes that perturbation of Light which is made below the Glasse In like manner that CGK shewes the perturbation of Light within the Glasse and CKL that which is below the Glasse From whence there are four divers motions or four different illuminations or Colours whose differences appear most manifestly to the Sense in a Prisma whose base is an equilaterall triangle when the Sunne-beames that passe through it are received upon a white paper For the triangle AFH appears Red to the Sense the triangle AHI Yellow the triangle CGK Green and approaching to Blew and lastly the triangle CKL appears Purple It is therefore evident that when weak but First Light passeth through a more resisting diaphanous Body as Glasse the beames which fall upon it tranversly make Rednesse and when the same First Light is stronger as it is in the thinner Medium below the straight line AC the transverse beames make Yellownesse Also when Second Light is strong