A B C D B K. 2 A N I B are proportionals And by taking the halfes of the third the fourth A B C D+B E. 2 A B C D A B I. A N I B are also proportionals which was to be proved 10 From what has been said of Deficient Figures described in a Parallelogram may be found out what proportions Spaces transmitted with accelerated Motion in determined times have to the times themselves according as the moved Body is accelerated in the several times with one or more degrees of Velocity For let the Parallelogram A B C D in the 6th figure and in it the three-sided figure D E B C be described and let F G be drawn any where parallel to the base cutting the Diagonal B D in H and the crooked line B E D in E let the proportion of B C to B F be for example triplicate to that of F G to F E whereupon the figure D E B C will be triple to its Complement B E D A and in like manner I F being drawn parallel to B C the three-sided figure E K B F will be triple to its Complement B K E I. Wherefore the parts of the Deficient Figure cut off from the Vertex by straight lines parallel to the base namely D E B C and E K B F will be to one another as the Parallelograms A C and I F that is in proportion compounded of the proportions of the altitudes and bases Seeing therefore the proportion of the altitude B C to the altitude B F is triplicate to the proportion of the base D C to the base F E the figure D E B C to the figure E K B F will be quadruplicate to the proportion of the same D C to F E. And by the same method may be found out what proportion any of the said three-sided figures has to any part of the same cut off from the Vertex by a straight line parallel to the base Now as the said figures are understood to be described by the continual decreasing of the base as of C D for example till it end in a point as in B so also they may be understood to be described by the continual encreasing of a point as of B till it acquire any magnitude as that of C D. Suppose now the figure B E D C to be described by the encreasing of the point B to the magnitude C D. Seeing therefore the propor ion of B C to B F is triplicate to that of C D to F E the proportion of F E to C D will by Conversion as I shall presently demonstrate be triplicate to that B F to B C. Wherefore if the straight line B C be taken for the measure of the time in which the point B is moved the Figure E K B F will represent the Sum of all the encreasing Velocities in the time B F and the figure D E B C will in like manner represent the Summe of all the encreasing Velocities in the time B C. Seeing therefore the proportion of the figure E K B F to the figure D E B C is compounded of the proportions of altitude to altitude and base to base and seeing the proportion of F E to C D is triplicate to that of B F to B C the proportion of the figure E K B F to the figure D E B C will be quadruplicate to that of B F to B C that is the proportion of the Sum of the Velocities in the time B F to the Sum of the Velocities in the time B C wil be quadruplicate to the proportion of B F to B C. Wherfore if a Body be moved from B with Velocity so encreasing that the Velocity acquired in the time B F be to the Velocity acquired in the time B C in triplicate proportion to that of the times themselves B F to B C and the Body be carried to F in the time B F the same Body in the time B C will be carried through a line equal to the fifth proportional from B F in the continual proportion of B F to B C. And by the same manner of working we may determine what Spaces are transmitted by Velocities encreasing according to any other proportions It remains that I demonstrate the proportion of F E to C D to be triplicate to that of B F to B C. Seeing therefore the proportion of C D that is of F G to F E is subtriplicate to that of B C to B F the proportion of F G to F E will also be subtriplicate to that of F G to F H. Wherefore the proportion of F G to F H is triplicate to that of F G that is of C D to F E. But in four continual proportionals of which the least is the first the proportion of the first to the fourth by the 16 Art of the 13 Chap. is subtriplicate to the proportion of the third to the same fourh Wherefore the proportion of F H to G F is subtriplicate to that of F E to C D and therefore the proportion of F E to C D is triplicate to that of F H to F G that is of B F to B C which was to be proved It may from hence be collected that when the Velocity of a Body is encreased in the same proportion with that of the times the degrees of Velocity above one another proceed as numbers do in immediate succession from Unity namely as 1 2 3 4 c. And when the Velocity is encreased in proportion duplicate to that of the times the degrees proceed as numbers from Unity skipping One as 1 3 5 7 c. Lastly when the proportions of the Velocities are triplicate to those of the times the progression of the degrees is as that of numbers from Unity skipping Two in every place as 1 4 7 10 c. and so of other proportions For Geometrical proportionals when they are taken in every point are the same with Arithmetical proportionals 11 Moreover it is to be noted that as in quantities which are made by any magnitudes decreasing the proportions of the figures to one another are as the proportions of the altitudes to those of the bases so also it is in those which are made with motion decreasing which motion is nothing else but that power by which the described figures are greater or less And therefore in the description of Archimedes his Spiral which is done by the continual diminution of the Semidiameter of a Circle in the same proportion in which the Circumference is diminished the Space which is contained within the Semidiameter and the Spiral Line is a third part of the whole Circle For the Semidiameters of Circles in as much as Circles are understood to be made up of the aggregate of them are so many Sectors and therefore in the description of a Spiral the Sector which describes it is diminished in duplicate proportions to the diminutions of the Circumference of the Circle in
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
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
is considered as a point or as in a Divided Body In an Undivided Body when we suppose the way by which the Motion is made to be a Line and in a Divided Body when we compute the Motion of the several parts of that Body as of Parts Secondly From the diversity of the regulation of Motion it is in a Body considered as Undivided sometimes Uniform and sometimes Multiform Uniform is that by which equal Lines are alwayes transmitted in equal times Multiform when in one time more in another time less space is transmitted Again of Multiform Motions there are some in which the degrees of Acceleration and Retardation proceed in the same proportions which the Spaces transmitted have whether duplicate or triplicate or by whatsoever number multiplyed and others in which it is otherwise Thirdly from the number of the Movents that is one Motion is made by one Movent onely and another by the concourse of many Movents Fourthly from the position of that Line in which a Body is moved in respect of some other Line and from hence one Motion is called Perpendicular another Oblique another Parallel Fifthly from the position of the Movent in respect of the Moved Body from whence one Motion is Pulsion or Driving another Traction or Drawing PULSION when the Movent makes the Moved Body goe before it and TRACTION when it makes it follow Again there are two sorts of Pulsion one when the motions of the Movent and Moved Body begin both together which may be called TRUSION or Thrusting and VECTION the other when the Movent is first moved and afterwards the Moved Body which Motion is called PERCUSSION or Stroke Sixthly Motion is considered sometimes from the Effect onely which the Movent works in the Moved Body which is usually called Moment Now MOMENT is the Excess of Motion which the Movent has above the Motion or Endeavour of the Resisting Body Seventhly it may be considered from the diversity of the Medium as one Motion may be made in Vacuity or empty Place another in a fluid another in a consistent Medium that is a Medium whose parts are by some power so consistent and cohering that no part of the same will yeild to the Movent unless the whole yeild also Eighthly when a Moved Body is considered as having parts there arises another distinction of Motion into Simple and Compounded Simple when all the several parts describe several equal lines Compounded when the lines described are Unequal 5 All Endeavour tends towards that part that is to say in that way which is determined by the Motion of the Movent if the Movent be but one or if there be many Movents in that way which their concourse determines For example if a Moved Body have direct Motion its first Endeavour will be in a Straight line if it have Circular Motion its first Endeavour will be in the Circumference of a Circle whatsoever the line be in which a Body has its Motion from the concourse of two Movents as soon as in any point thereof the force of one of the Movents ceases there immediately the former Endeavour of that Body will be changed into an Endeavour in the line of the other Movent 6 Wherefore when any Body is carried on by the concourse of two Winds one of those Winds ceasing the Endeavour and Motion of that Body will be in that line in which it would have been carried by that Wind alone which blows still And in the describing of a Circle where that which is moved has its Motion determined by a Movent in a Tangent and by the Radius which keeps it in a certain distance from the Center if the retention of the Radius cease that Endeavour which was in the Circumference of the Circle will now be in the Tangent that is in a Straight line For seeing Endeavour is computed in a lesse part of the Circumference then can be given that is in a point the way by which a Body is moved in the Circumference is compounded of innumerable Straight lines of which every one is less then can be given which are therefore called Points Wherefore when any Body which is moved in the Circumference of a Circle is freed from the retention of the Radius it will proceed in one of those Straight lines that is in a Tangent 7 All Endeavour whether strong or weak is propagated to infinite distance for it is Motion If therefore the first Endeavour of a Body be made in Space which is empty it will alwayes proceed with the same Velocity for it cannot be supposed that it can receive any resistance at all from empty Space and therefore by the 7 Article of the 9 Chapter it will alwayes proceed in the same way and with the same Swiftness And if its Endeavour be in Space which is filled yet seeing Endeavour is Motion that which stands next in its way shall be removed and endeavour further and again remove that which stands next so infinitely Wherefore the propagation of Endeavour from one part of full Space to another proceeds infinitely Besides it reaches in any instant to any distance how great soever For in the same instant in which the first part of the full Medium removes that which is next it the second also removes that part which is next to it and therefore all Endeavour whether it be in empty or in full Space proceeds not onely to any distance how great soever but also in any time how little soever that is in an instant Nor makes it any matter that Endeavour by proceeding growes weaker and weaker till at last it can no longer be perceived by Sense for Motion may be insensible and I do not here examine things by Sense and Experience but by Reason 8 When two Movents are of equal Magnitude the swifter of them works with greater force then the slower upon a Body that resists their Motion Also if two Movents have equal Velocity the greater of them works with more force then the less For where the Magnitude is equal the Movent of greater Velocity makes the greater impression upon that Body upon which it falls and where the Velocity is equal the Movent of greater Magnitude falling upon the same point or an equal part of another Body loses less of its Velocity because the resisting Body works onely upon that part of the Movent which it touches and therefore abates the Impetus of that part onely whereas in the mean time the parts which are not touched proceed and retein their whole force till they also come to be touched and their force has some effect Wherfore for example in Batteries a longer then a shorter piece of Timber of the same thickness and velocity and a thicker then a slenderer piece of the same length and velocity works a greater effect upon the Wall CHAP. XVI Of Motion Accelerated and Vniform and of Motion by Concourse 1 The Velocity of any Body in what Time
of those two Movents the Body will be carried through the Semipabolical crooked line A G D. For let the parallelelogram A B D C be completed from the point E taken any where in the straight line A B let E F be drawn parallel to A C and cutting the crooked line in G and lastly through the point G let A I be drawn parallel to the straight lines A B and C D. Seeing therefore the proportion of A B to A E is by supposition duplicate to the proportion of E F to E G that is of the time A C to the time A H at the same time when A C is in E F A B will be in H I and therefore the moved Body will be in the common point G. And so it will alwayes be in what part soever of A B the point E be taken Wherefore the moved Body will always be found in the parabolical line A G D which was to be demonstrâ—ted 10 If a Body be carried by two Movents together which meet in any given angle and are moved the one Uniformly the other with Impetus encreasing from Rest till it be equal to that of the Uniform Motion and with such acceleration that the proportion of the Lengths transmitted be every where triplicate to that of the Times in which they are transmitted The line in which that Body is moved will be the crooked line of the first Semiparabolaster of two Means whose baâ—e is the Impetus last acquired Let the straight line A B in the 6th Figure be moved Uniformly to C D and let another Movent A C be moved at the same time to B D with motion so accelerated that the proportion of the Lengths transmitted by every where triplicate to the proportion of their Times and let the Impetus acquired in the end of that motion be B D equal to the straight line A C lastly let A D be the crooked line of the first Semiparabolaster of two Means I say that by the concourse of the two Movents together the Body will be alwayes in that crooked line A D. For let the parallelogram A B D C be completed and from the point E taken any where in the straight line A B let E F be drawn parallel to A C and cutting the crooked line in G and through the point G let H I be drawn parallel to the straight lines A B and C D. Seeing therefore the proportion of A B to A E is by supposition triplicate to the proportion of E F to E G that is of the time A C to the time A H at the same time when A C is in E F A B will be in H I and therefore the moved Body will be in the common point G. And so it will alwayes be in what part soever of A B the point E be taken and by consequent the Body will always be in the crooked line A G D which was to be demonstrated 11 By the same method it may be shewn what line it is that it made by the motion of a Body carried by the concourse of any two Movents which are moved one of them Uniformly the other with acceleration but in such proportions of Spaces and Times as are explicable by Numbers as duplicate triplicate c. or such as may be designed by any broken number whatsoever For which this is the Rule Let the two numbers of the Length Time be added together let their Sum be the Denominator of a Fraction whose Numerator must be the number of the Length Seek this Fraction in the Table of the third Article of the 17th Chapter and the line sought will be that which denominates the three-sided Figure noted on the left hand and the kind of it will be that which is numbred above over the Fraction For example Let there be a concourse of two Movements whereof one is moved Uniformly the other with motion so accelerated that the Spaces are to the Times as 5 to 3. Let a Fraction be made whose Denominator is the Sum of 5 and 3 and the Numerator 5 namely the Fraction â… Seek in the Table and you will find â… to be the third in that row which belongs to the three-sided Figure of four Means Wherfore the line of Motion made by the concourse of two such Movents as are last of all described will be the crooked line of the third Parabolaster of four Means 12 If Motion be made by the concourse of two Movents whereof one is moved Uniformly the other beginning from Rest in the Angle of concourse with any acceleration whatsoever the Movent which is Moved Uniformly shall put forward the moved Body in the several parallel Spaces lesse then if both the Movents had Uniform motion and still lesse and lesse as the Motion of the other Movent is more and more accelerated Let the Body be placed in A in the 7th figure and be moved by two Movents by one with Uniform Motion from the straight line A B to the straight line C D parallel to it and by the other with any acceleration from the straight line A C to the straight line B D parallel to it and in the parallelogram A B D C let a Space be taken between any two parallels E F and G H. I say that whilest the Movent A C passes through the latitude which is between E F and G H the Body is lesse moved forwards from A B towards C D then it would have been if the Motion from A C to B D had been Uniform For suppose that whilest the Body is made to descend to the parallel E F by the power of the Movent from A C towards B D the same Body in the same time is moved forwards to any point F in the line E F by the power of the Movent from A B towards C D and let the straight line A F be drawn and produced indeterminately cutting G H in H. Seeing therefore it is as A E to A G so E F to G H if A C should descend towards B D with uniform Motion the Body in the time G H for I make A C and its parallels the measure of time would be found in the point H. But because A C is supposed to be moued towards B D which motion continually accelerated that is in greater proportion of Space to Space then of Time to Time in the time G H the Body will be in some parallel beyond it as between G H and B D. Suppose now that in the end of the time G H it be in the parallel I K in I K let I L be taken equal to G H. When therefore the Body is in the parallel I K it will be in the point L. Wherefore when it was in the parallel G H it was in some point between G and H as in the point M but if both the Motions had been Uniform it had been in the point H and therefore whilest the Movent
which it is inscribed so that the Complement of the Spiral that is that space in the Circle which is without the Spiral Line is double to the space within the Spiral Line In the same manner if there be taken a mean proportional every where between the Semidiameter of the Circle which contains the Spiral and that part of the Semidiameter which is within the same there will be made another figure which will be half the Circle And to conclude this Rule serves for all such Spaces as may be described by a Line or Superficies decreasing either in magnitude or power so that if the proportions in which they decrease be commensurable to the proportions of the times in which they decrease the magnitudes of the figures they describe will be known 12 The truth of that proposition which I demonstrated in the second Article which is the foundation of all that has been said concerning Deficient Figures may be derived from the Elements of Philosophy as having iâ—â— original in this That all equality and inequality between two effects that is all Proportion proceeds from and is determined by the equal and unequal causes of those effects or from the proportion which the causes concurring to one effect have to the causes which concurre to the producing of the other effect and that therefore the proportions of Quantities are the same with the proportions of their causes Seeing therefore two Deficient Figures of which one is the Complement of the other are made one by motion decreasing in a certain time and proportion the other by the loss of Motion in the same time the causes which make and determine the quantities of both the figures so that they can be no other then they are differ onely in this that the proportions by which the quantity which generates the figure proceeds in describing of the same that is the proportions of the remainders of all the times and altitudes may be other proportions then those by which the same generating quantity decreases in making the Complement of that Figure that is the proportions of the quantity which generates the Figure continually diminished Wherefore as the proportions of the quantity in which Motion is lost is to that of the decreasing quantities by which the Deficient Figure is generated so will the Defect or Complement be to the Figure it self which is generated 13 There are also other quantities which are determinable from the knowledge of their causes namely from the comparison of the Motions by which they are made and that more easily then from the common Elements of Geometry For example That the Superficies of any portion of a Sphere is equal to that Circle whose Radius is a straight Line drawn from the Pole of the portion to the Circumference of its base I may demonstrate in this manner Let B A C in the 7 Figure be a portion of a Sphere whose Axis is A E whose base is B C let A B be the straight line drawn from the Pole A to the base in B and let A D equal to A B touch the great Circle B A C in the Pole A. It is to be proved that the Circle made by the Radius A D is equal to the Superficies of the portion B A C. Let the plain A E B D be understood to make a revolution about the Axis A E it is manifest that by the straight line A D a Circle will be described and by the arch A B the Superficies of a portion of a Sphere and lastly by the Subtense A B the Superficies of a right Cone Now seeing both the straight line A B and the arch A B make one and the same revolution and both of them have the same extreme points A and B the cause why the the Spherical Superficies which is made by the arch is greater then the Conical Superficies which is made by the Subtense is that A B the arch is greater then A B the Subtense and the cause why it is greater consists in this that although they be both drawn from A to B yet the Subtense is drawn straight but the arch angularly namely according to that angle which the arch makes with the Subtense which angle is equal to the angle D A B for an angle of contingence adds nothing to an angle of a Segment as has been shewn in the 14 Chapter at the 16th Article Wherefore the magnitude of the angle D A B is the cause why the Superficies of the portion described by the arch A B is greater then the Superficies of the right Cone described by the Subtense A B. Again the cause why the Circle described by the Tangent A D is greater then the Superficies of the right Cone described by the Subtense A B notwitstanding that the Tangent and the Subtense are equal and both moved round in the same time is this that A D stands at right angles to the Axis but A B obliquely which obliquity consists in the same angle D A B. Seeing therefore the quantity of the angle D A B is that which makes the excess both of the Superficies of the Portion and of the Circle made by the Radius A D above the superficies of the Right Cone described by the subtense A B it follows that both the Superficies of the Portion and that of the Circle do equally exceed the Superficies of the Cone Wherefore the Circle made by A D or A B and the Spherical Superficies made by the arch A B are equal to one another which was to be proved â—4 If these Deficient Figures which I have described in a 〈◊〉 were capable of exact description then any number of mean proportionals might be found out between two straight lines given For example in the Parallelogram A B C D in the 8th Figure let the three-sided figure of two Means be described which many call a Cubical Parabola and let R and S be two given straight lines between which if it be required to find two mean proportionals it may be done thus Let it be as R to S so B C to B F and let F E be drawn parallel to B A and cut the crooked line in E then through E let G H be drawn parallel and equal to the straight line A D and cut the Diagonal B D in I for thus we have G I the greatest of two Means between G H and G E as appears by the description of the figure in the 4th Article Wherefore if it be as G H to G I so R to another line T that T will be the greatest of two Means between R and S. And therefore if it be again as R to T so T to another line X that will be done which was required In the same manner four mean proportionals may be found out by the description of a three-sided figure of four Means and so any other number of Means c. CHAP. XVIII Of the Equation of Straight Lines with the
drawn from the propositions which prove the same But the cause of his construction is in the things themselves and consists in motion or in the concourse of motions Wherefore those propositions in which Analysis ends are Definitions but such as signifie in what manner the construction or generation of the thing proceeds For otherwise when he goes back by Synthesis to the proofe of his Probleme he will come to no Demonstration at all there being no true Demonstration but such as is scientificall and no Demonstration is scientifical but that which proceeds from the knowledge of the causes from which the construction of the Probleme is drawne To collect therefore what has been said into few words ANALYSIS is Ratiocination from the supposed construction or generation of a thing to the efficient cause or coefficient causes of that which is constructed or generated And SYNTHESIS is Ratiocination from the first causes of the Construction continued through all the middle causes till we come to the thing it selfe which is constructed or generated But because there are many means by which the same thing may be generated or the same Probleme be constructed therefore neither do all Geometricians nor doth the same Geometrician alwayes use one and the same Method For if to a certain quantity given it be required to construct another quantity equal there may be some that will enquire whether this may not be done by means of some motion For there are quantities whose equality and inequality may be argued from Motion and Time as well as from Congruence and there is motion by which two quantities whether Lines or Superficies though one of them be crooked the other straight may be made congruous or coincident And this method Archimedes made use of in his Book de Spiralibus Also the equality or inequality of two quantities may be found out and demonstrated from the consideration of Waight as the same Archimedes did in his Quadrature of the Parabola Besides equality and equality are found out often by the division of the two quantityes into parts which are considered as undivisible as Cavallerius Bonaventura has done in our time and Archimedes often Lastly the same is performed by the consideration of the Powers of lines or the roots of those Powers and by the multiplication division addition and substraction as also by the extraction of the roots of those Powers or by finding where straight lines of the same proportion terminate For example when any number of straight lines how many soever are drawne from a straight line and passe all through the same point looke what proportion they have and if their parts continued from the point retaine every where the same proportion they shall all terminate in a straight line And the same happens if the point be taken between two Circles So that the places of all their points of termination make either straight lines or circumferences of Circles and are called Plain Places So also when straight parallel lines are applyed to one straight line if the parts of the straight line to which they are applyed be to one another in proportion duplicate to that of the contiguous applyed lines they will all terminate in a Conical Section which Section being the place of their termination is called a Solid Place because it serves for the finding out of the quantity of any Equation which consists of three dimensions There are therfore three ways of finding out the cause of Equality or Inequality between two given quantities namely First by the Computation of Motions for by equal Motion equal Time equal Spaces are described and Ponderation is motion Secondly By Indivisibles because all the parts together taken are equal to the whole And thirdly by the Powers for when they are equall their roots also are equall and contrarily the Powers are equall when their roots are equal But if the question be much complicated there cannot by any of these wayes be constituted a certaine Rule from the supposition of which of the unknown quantities the Analysis may best begin nor out of the variety of Equations that at first appeare which we were best to choose but the successe will depend upon dexterity upon formerly acquired Science and many times upon fortune For no man can ever be a good Analyst without being first a good Geometrician nor do the rules of Analysis make a Geometrician as Synthesis doth which begins at the very Elements and proceeds by a Logical Use of the same For the true teaching of Geometry is by Synthesis according to Euclides method and he that hath Euclide for his Master may be a Geometrician without Vieta though Vieta was a most admirable Geometrician but he that has Vieta for his master not so without Euclide And as for that part of Analysis which works by the Powers though it be esteemed by some Geometricians not the chiefest to be the best way of solving all Problemes yet it is a thing of no great extent it being all contained in the doctrine of rectangles and rectangled Solids So that although they come to an Equation which determines the quantity sought yet they cannot sometimes by art exhibit that quantity in a Plain but in some Conique Section that is as Geometricians say not Geometrically but mechanically Now such Problemes as these they call Solid and when they cannot exhibit the quantity sought for with the helpe of a conique Section they call it a Lineary Probleme And therefore in the quantities of angles and of the arches of Circles there is no use at all of the Analyticks which proceed by the Powers so that the Antients pronounced it impossible to exhibit in a plaine the Division of Angles except bisection and the bisection of the bisected parts otherwise then mechanically For Pappus before the 31 proposition of his fourth Book distinguishing and defining the several kinds of Problemes says that some are Plain others Solid and others Lineary Those therefore which may be solved by straight lines and the circumferences of Circles that is which may be described with the Rule and Compass without any other Instrument are fitly called Plain for the lines by which such Problemes are found out have their generation in a Plain But those which are solved by the using of some one or more Conique Sections in their construction are called Solid because their construction cannot be made without using the superficies of solid figures namely of Cones There remains the third kinde which is called Lineary because other lines besides those already mentioned are made use of in their construction c. And a little after he sayes Of this kinde are the Spiral lines the Quadratrices the Conchoeides and the Cissoeides And Geometricians think it no small fault when for the finding out of a Plain Probleme any man makes use of Coniques or new Lines Now he ranks the Trisection of an angle among Solid Problemes and the Quinquesection among Lineary But what are the ancient Geometricians
to it self so that in what part soever of the Ecliptick the Center of the Epicycle be found and in what part soever of the Epicycle the Center of the Earth be found at the same time the Axis of the Earth will be parallel to the place where the same Axis would have been if the Center of the Earth had never gone out of the Ecliptick Now as I have demonstrated the simple annual motion of the Earth from the supposition of Simple Motion in the Sunne so from the supposition of Simple Motion in the Earth may be demonstrated the monethly Simple Motion of the Moon For if the names be but changed the Demonstration will be the same and therefore need not be repeated 7 That which makes this supposition of the Sunnes Simple Motion in the Epicycle fghi probable is First that the Periods of all the Planets are not onely described about the Sunne but so described as that they are al contained within the Zodiack that is to say within the latitude of about 16 degrees for the cause of this seems to depend upon some power in the Sunne especially in that part of the Sunne which respects the Zodiack Secondly that in the whole coâ—passe of the heavens there appears no other Body from which the cause of this Phaenomenon can in probability be derived Besides I could not imagine that so many and such various motions of the Planets should have no dependance at all upon one another But by supposing motive power in the Sunne we suppose motion also for power to move without motion is no power at all I have therefore supposed that there is in the Sunne for the governing of the primary Planets and in the Earth for the governing of the Moon such motion as being received by the primary Planets and by the Moon makes them necessarily appear to us in such manner as we see them Whereas that circular motion which is commonly attributed to them about a fixed Axis which is called Conversion being a motion of their parts onely and not of their whole Bodies is insufficient to salve their Appearances For seeing whatsoever is so moved hath no endeavour at all towards those parts which are without the circle they have no power to propagate any endeavour to such Bodies as are placed without it And as for them that suppose this may be done by Magnetical Virtue or by incorporeall and immateriall Species they suppose no naturall cause nay no cause at all For there is no such thing as an Incorporeal Movent and Magnetical Virtue is a thing altogether unknown and whensoever it shall be known it will be found to be a motion of Body It remaines therefore that if the primary Planets be carried about by the Sunne and the Moon by the Earth they have the simple circular motions of the Sunne and the Earth for the causes of their circulations Otherwise if they be not carried about by the Sunne and the Earth but that every Planet hath been moved as it is now moved ever since it was made there will be of their motions no cause naturall For either these motions were concreated with their Bodies and their cause is supernatural or they are coeternal with them and so they have no cause at all For whatsoever is Eternall was never generated I may add besides to confirme the probability of this simple motion that allmost all learned men are now of the same opinion with Copernicus concerning the parallelisme of the Axis of the Earth it seemed to me to be more agreeable to truth or at least more handsome that it should be caused by simple Circular Motion alone than by two motions one in the Ecliptick and the other about the Earths own Axis the contrary way neither of them Simple nor either of them such as might be produced by any motion of the Sunne I thought best therefore to retain this Hypothesis of Simple Motion and from it to derive the causes of as many of the Phaenomena as I could and to let such alone as I could not deduce frm thence It will perhaps be objected that although by this supposition the reason may be given of the Parallelisme of the Axis of the Earth and of many other Appearances nevertheless seeing it is done by placing the Body of the Sunne in the Center of that Orbe which the Earth describes with its annual motion the supposition it self is false because this annual Orbe is excentrique to the Sunne In the first place therefore let us examine what that Excentricity is and whence it proceeds 8 Let the annual Circle of the Earth abcd in the same 3d figure be divided into four equal parts by the straight lines ac bd cutting one another in the Center e and let a be the beginning of Libra b of Capricorn c of Aries and d of Cancer and let the whole Orbe abcd be understood according to Copernicus to have every way so great distance from the Zodiack of the fixed Starres that it be in comparison with it but as a point Let the Earth be now supposed to be in the beginning of Libra at a. The Sunne therefore will appear in the beginning of Aries at c. Wherefore if the Earth be moved from a to b the apparent motion of the Sunne will be from c to the beginning of Cancer in d and the Earth being moved forwards from b to c the Sunne also will appear to be moved forwards to the beginning of Libra in a Wherefore cda will be the Summer Arch and the Winter Arch will be abc Now in the time of the Suns apparent motion in the Summer Arch there are numbred 186¾ dayes and consequently the Earth makes in the same time the same number of diurnal conversions in the Arch abc and therefore the Earth in its motion through the Arch cda will make onely 178½ diurnal conversions Wherefore the Arch a b c ought to be greater then the Arch c d a by 8¼ dayes that is to say by almost so many degrees Let the Arch a r as also c s be each of them an Arch of two degrees and 1 16. Wherefore the Arch r b s will be greater then the Semicircle a b c by 4 degrees and â…› and greater then the Arch s d r by 8 degrees and ¼ The Equinoxes therefore will be in the points r s and therefore also when the Earth is in r the Sunne will appear in s. Wherefore the true place of the Sunne will be in t that is to say without the Center of the Earths annual motion by the quantity of the Sine of the Arch a r or the Sine of two degrees and 16 minutes Now this Sine putting 100000 for the Radius will be neer 3580 parts thereof And so munh is the Excentricity of the Earths annual motion provided that that motion be in a perfect circle and s r are the Equinoctial points and the straight lines s r c a produced both wayes till they
of the parts of those plants made an Odorous liquour so also of aire passing through the same plants whilest they are growing are made Odorous aires And thus also it is with the Juices and Spirits which are bred in Living Creatures 16 That Odorous Bodies may be made more Odorous by Contrition proceeds from this that being broken into many parts which are all Odorous the aire which by respiration is drawn from the Object towards the Organ doth in its passage touch upon all those parts and receives their motion Now the aire toucheth the superficies onely and a Body having less superficies whilest it is whole then all its parts together have after it is reduced to powder it follows that the same Odorous Body yeildeth less Smell whilest it is whole then it will do after it is broken into smaller parts And thus much of Smels 17 The Tast follows whose generation hath this difference from that of the Sight Hearing and Smelling that by these we have Sense of remote Objects whereas we Tast nothing but what is contiguous and doth immediately touch either the Tongue or Palate or both From whence it is evident that the cuticles of the Tongue and Palate and the Nerves inserted into them are the first Organ of Tast and because from the concussion of the parts of these there followeth necessarily a concussion of the Pia Mater that the action communicated to these is propagated to the Brain and from thence to the farthest Organ namely the Heart in whose reaction consisteth the nature of Sense Now that Savours as well as Odours doe not onely move the Brain but the Stomack also as is manifest by the loathings that are caused by them both they that consider the Organ of both these Senses will not wonder at all seeing the Tongue the Palate the Nostrils have one and the same continued cuticle derived from the Dura Mater And that Effluviums have nothing to doe in the Sense of Tasting is manifest from this that there is no Tast where the Organ and the Object are not contiguous By what variety of motions the different kinds of Tasts which are innumerable may be distinguished I know not I might with others derive them from the divers figures of those Atomes of which whatsoever may be Tasted consisteth or from the diverse motions which I might by way of Supposition attribute to those Atomes conjecturing not without some likelyhood of truth that such things as tast Sweet have their particles moved with slow circular motion and their figures Spherical which makes them smooth and pleasing to the Organ that Bitter things have circular motion but vehement and their figures full of Angles by which they trouble the Organ and that Sowre things have straight and reciprocal motion and their figures long and small so that they cut and wound the Organ And in like manner I might assigne for the causes of other Tasts such several motions and figures of Atomes as might in probability seem to be the true causes But this would be to revolt from Philosophy to Divination 18 By the Touch we feel what Bodies are Cold or Hot though they be distant from us Others as Hard Soft Rough and Smooth we cannot feel unless they be contiguous The Organ of Touch is every one of those membranes which being continued from the Pia Mater are so diffused throughout the whole Body as that no part of it can be pressed but the Pia Mater is pressed together with it Whatsoever therefore presseth it is felt as Hard or Soft that is to say as more or less Hard. And as for the Sense of Rough it is nothing else but innumerable perceptions of Hard and Hard succeeding one another by short intervals both of time and place For we take notice of Rough and Smooth as also of Magnitude and Figure not onely by the Touch but also by Memory For though some things are touched in one Point yet Rough and Smooth like Quantity and Figure are not perceived but by the Flux of a Point that is to say we have no Sense of them without Time and we can have no Sense of Time without Memory CHAP. XXX Of Gravity 1 A Thick Body doth not contain more Matter unless also more Place then a Thinne 2 That the Descent of Heavy Bodies proceeds not from their own Appetite but from some Power of the Earth 3 The difference of Gravities proceedeth from the difference of the Impetus with which the Elements whereof Heavy Bodies are made do fall vpon the Earth 4 The cause of the Descent of Heavy Bodies 5 In what proportion the Descent of Heavy Bodies is accelerated 6 Why those that Dive do not when they are under Water feel the waight of the Water above them 7 The Waight of a Body that floateth is equal to the Waight of so much Water as would fill the space which the immersed part of the Body takes up within the Water 8 If a Body be Lighter then Water then how big soever that Body be it will float upon any quantity of Water how little soever 9 How Water may be lifted up and forced out of a Vessel by Air. 10 Why a Bladder is Heavier when blown full of aire then when it is empty 11 The cause of the ejection upwards of Heavy Bodies from a Wind-Gun 12 The cause of the ascent of Water in a Weather-glass 13 The cause of motion upwards in Living Creatures 14 That there is in Nature a kind of Body Heavier then Aire which nevertheless is not by Sense distinguishable from it 15 Of the cause of Magnetical vertue 1 IN the 21 Chapter I have defined Thick and Thinne as that place required so as that by Thick was signified a more Resisting Body and by Thinne a Body less Resisting following the custome of those that have before me discoursed of Refraction Now if we consider the true and vulgar signification of those words we shall find them to be Names Collective that is to say Names of Multitude as Thick to be that which takes up more parts of a space given Thinne that which contains fewer parts of the same magnitude in the same space or in a space equal to it Thick therefore is the same with Frequent as a Thick Troop And Thinne the same with Unfrequent as a Thinne Rank Thinne of Houses not that there is more matter in one place then in another equal place but a greater quantity of some named Body For there is not less matter or Body indefinitely taken in a Desert then there is in a City but fewer Houses or fewer Men. Nor is there in a Thick Rank a greater quantity of Body but a greater number of Souldiers then in a Thinne Wherefore the multitude paucity of the parts contained within the same space do constitute Density and Rarity whether those parts be separated by Vacuum or by Aire But the consideration of this is not of any great moment in Philosophy and therefore I let
in like manner is followed by the noxious matter contained in CB by this means the pit is for that time made healthful Out of this History which I write onely to such as have had experience of the truth of it without any designe to support my Philosophy with Stories of doubtful credit may be collected the following possible cause of this Phaenomenon namely that there is a certain matter fluid most transparent and not much lighter then water which breaking out of the Earth fills the Pit to C and that in this matter as in water both Fire and Living creatures are extinguished 15 About the nature of Heavy Bodies the greatest difficulty ariseth from the contemplation of those things which make other Heavy Bodies ascend to them such are Jet Amber and the Loadstone But that which troubles men most is the Loadstone which is also called Lapis Herculeus a stone though otherwise despicable yet of so great power that it taketh up Iron from the Earth and holds it suspended in the aire as Hercules did Antaeus Nevertheless we wonder at it somewhat the less because we see Jet draw up Straws which are Heavy Bodies though not so Heavy as Iron But as for Jet it must first be excited by rubbing that is to say by motion to and fro whereas the Loadstone hath sufficient excitation from its own nature that is to say from some internal principle of motion peculiar to it self Now whatsoever is moved is moved by some contiguous and moved Body as hath been formerly demonstrated And from hence it follows evidently that the first endeavour which Iron hath towards the Loadstone is caused by the motion of that aire which is contiguous to the Iron Also that this motion is generated by the motion of the next aire and so on successively till by this succession we find that the motion of all the intermediate air taketh its beginning from some motion which is in the Loadstone it self which motion because the Loadstone seems to be at rest is invisible It is therefore certain that the attractive power of the Loadstone is nothing else but some motiō of the smallest particles thereof Supposing therefore that those small Bodies of which the Loadstone is in the bowels of the Earth composed have by nature such motion or endeavour as was above attributed to Jet namely a reciprocal motiō in a line too short to be seen both those stones wil have one the same cause of attraction Now in what manner and in what order of working this cause produceth the effect of attraction is the thing to be enquired And first we know that when the string of a Lute or Viol is stricken the Vibration that is the reciprocal motion of that string in the same straight Line causeth like Vibration in another string which has like tension We know also that the dregs or small sands which sink to the bottom of a Vessel will be raised up from the bottom by any strong and reciprocal agitation of the water stirred with the hand or with a staff Why therefore should not reciprocal motion of the parts of the Loadstone contribute as much towards the moving of Iron For if in the Loadstone there be supposed such reciprocal motion or motion of the parts forwards and backwards it will follow that the like motion will be propagated by the aire to the Iron and consequently that there will be in all the parts of the Iron the same reciprocations or motions forwards and backwards And from hence also it will follow that the intermediate aire between the Stone and the Iron will by little and little be thrust away and the aire being thrust away the Bodies of the Loadstone and the Iron will necessarily come together The possible cause therefore why the Loadstone and Jet draw to them the one Iron the other Strawes may be this that those attracting Bodies have reciprocal motion either in a straight line or in an Elliptical line when there is nothing in the nature of the attracted Bodies which is repugnant to such a motion But why the Loadstone if with the help of Cork it float at liberty upon the top of the water should from any position whatsoever so place it self in the plain of the Meridian as that the same points which at one time of its being at rest respect the Poles of the Earth should at all other times respect the same Poles the cause may be this That the reciprocal motion which I supposed to be in the parts of the Stone is made in a line parallel to the Axis of the Earth and has been in those parts ever since the Stone was generated Seeing therefore the Stone whilest it remains in the Mine and is carried about together with the Earth by its diurnal motion doth by length of time get a habit of being moved in a line which is perpendicular to the line of its reciprocal motion it will afterwards though its axis be removed from the parallel situation it had with the axis of the Earth retain its endeavour of returning to that situation again and all endeavour being the beginning of motion and nothing intervening that may hinder the same the Loadstone will therefore return to its former situation For any piece of Iron that has for a long time rested in the plain of the Meridian whensoever it is forced from that situation and afterwards left to its own liberty again will of it self return to lie in the Meridian again which return is caused by the endeavour it acquired from the diurnal motion of the Earth in the parallel circles which are perpendicular to the Meridians If Iron be rubbed by the Loadstone drawn from one Pole to the other two things will happen one that the Iron will acquire the same direction with the Loadstone that is to say that it will lie in the Meridian and have its Axis and Poles in the same position with those of the Stone the other that the like Poles of the Stone and of the Iron will avoid one another and the unlike Poles approach one another And the cause of the former may be this that Iron being touched by motion which is not reciprocal but drawn the same way from Pole to Pole there will be imprinted in the Iron also an endeavour from the same Pole to the same Pole For seeing the Loadstone differs from Iron no otherwise then as Ore from Metal there will be no repugnance at all in the Iron to receive the same motion which is in the Stone From whence it follows that seeing they are both affected alike by the diurnal motion of the Earth they will both equally return to their situation in the Meridian whensoever they are put frō the same Also of the later this may be the cause that as the Loadstone in touching the Irō doth by its action imprint in the Iron an endeavour towards one of the Poles suppose towards the North Pole so reciprocally the