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
compared with it according to Quantity Secondly seeing the external Angle made by a Subtense produced and the next Subtense is equal to an Angle from the Center insisting upon the same Arch as in the last figure the Angle GCD is equal to the Angle CAD the Angle of Contingence wil be equal to that Angle from the Center which is made by AB and the same AB for no part of a Tangent can subtend any Arch but as the point of Contact is to be taken for the Subtense so the Angle of Contingence is to be accounted for the external Angle and equal to that Angle whose Arch is the same point B. Now seeing an Angle in general is defined to be the Opening oâ— Divergence of two lines which concurre in one sole point seeing one Opening is greater then another it caÌ„not be denied but that by the very generation of it an Angle of Contingence is Quantity for wheresoever there is Greater and Less there is also Quantity but this quantity consists in greater and less Flexion for how much the greater a Circle is so much the neerer comes the Circumference of it to the nature of a straight Line for the Circumference of a Circle being made by the curvation of a straight line the less that straight line is the greater is the curvation therfore when one straight line is a Tangent to many Circles the Angle of Contingence which it makes with a less Circle is greater then that which it makes with a greater Circle Nothing therefore is added to or taken from an Angle simply so called by the addition to it or taking from it of never so many Angles of Contingence And as an Angle of one sort can never be equal to an Angle of the other sort so they cannot be either greater or less then one another From whence it follows that an Angle of a Segment that is the Angle which any straight line makes with any Arch is equal to the Angle which is made by the same straight line another which touches the Circle in the point of their Concurrence as in the last figure the Angle which is made between GB and BK is equal to that which is made between GB and the Arch BC. 17 An Angle which is made by two Plains is commonly called the Inclination of those Plains And because Plains have equal Inclination in all their parts instead of their Inclination an Angle is taken which is made by two straight lines one of which is in one the other in the other of those Plains but both perpendicular to the common Section 18 A Solid Angle may be conceived two wayes First for the aggregate of all the Angles which are made by the motion of a straight line while one extreme point thereof remayning fixed it is carried about any plain figure in which the fixed point of the straight line is not conteined And in this sense it seems to be understood by Euclide Now it is manifest that the quantity of a Solid Angle so conceived is no other then the aggregate of all the Angles in a Superficies so described that is in the Superficies of a Pyramidal Solid Secondly when a Pyâ—amis or Cone has its Vertex in the Center of a Sphere a Solid Angle may be understood to be the proportion of a Spherical Superficies subtending that Vertex to the whole Superficies of the Sphere In which sense solid Angles are to one another as the Spherical Bases of Solids which have their Vertex in the Center of the same Sphere 19 All the wayeâ— by which two lines respect one another or all the variety of their position may be comprehended under four kindes For any two lines whatsoever are either Parallels or being produced if need be or moved one of them to the other parallelly to it self they make an Angle or else by the like production and motion they Touch one another or lastly they are Asymptotes The nature of Parallels Angles and Tangents has been already declared It remains that I speak briefly of the nature of Asymptotes Asymptosy depends upon this that Quantity is infinitly divisible And from hence it follows that any line being given and a Body supposed to be moved from one extreme thereof towards the other it is possible by taking degrees of Velocity alwayes lesse and lesse in such proportion as the parts of the Line are made lesse by continual division that the same Body may be alwayes moved forwards in that Line and yet never reach the end of it For it is manifest that if any straight Line as AF in the 8th figure be cut any where in B and again BF be cut in C and CF in D and DF in E and so eternally and there be drawn from the point F the straight Line FF at any Angle AFF and lastly if the straight Lines AF BF CF DF EF c. having the same proportion to one another with the Segments of the Line AF be set in order and parallel to the same AF the crooked Line ABCDE and the straight Line FF will be Asymptotes that is they will alwayes come neerer and neerer together but never touch one another Now because any Line may be cut eternally according to the proportions which the Segments have to one another therefore the divers kindes of Asymptotes are infinite in number and not necessary to be further spoken of in this place In the nature of Asymptotes in general there is no more then that they come still neerer and neerer but never touch But in special in the Asymptosie of Hyperbolique Lines it is understood they should approach to a distance lesse then any given quantity 20 SITUATION is the relation of one place to another where there are many places their Situation is determined by four things By their Distances from one another By several Distances from a place assigned By the order of straight lines drawn from a place assigned to the places of them all and by the Angles which are made by the lines so drawn For if their Distances Order and Angles be given that is be certainly known their several places will also be so certainly known as that they can be no other 21 Points how many soever they be have Like Situation with an equal number of other Points when all the straight lines that are drawn from some one point to all these have severally the same proportion to those that are drawn in the same order and at equal Angles from some one point to all those For let there be any number of Points as A B and C in the 9 figure to which from some one point D let the straight Lines DA DB and DC be drawn and let there be an equal number of other Points as E F and G and from some point H let the straight Lines HE HF and HG be drawn so that the Angles ADB and BDC be severally and in the same order equal to the Angles EHF and
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 touches a Spiral at the end of its first conversion For upon the center A in the sixth figure let the circle BCDE be described and in it let Archimedes his Spiral AFGHB be drawn beginning at A and ending at
B. Through the center A let the straight line CE be drawn cutting the Diameter BD at right angles and let it be produced to I so that AI be equal to the Perimeter BCDEB Therefore IB being drawn will touch the Spiral AFGHB in B which is demonstrated by Archimedes in his book de Spiralibus And for a Straight Line equal to the given Spiral AFGHB it may be found thus Let the straight line AI which is equal to the Perimeter BCDE be bisected in K and taking KL equal to the Radius AB let the rectangle IL be completed Let ML be understood to be the axis and KL the base of a Parabola and let MK be the crooked line thereof Now if the point M be conceived to be so moved by the concourse of two movents the one froÌ„ IM to KL with velocity encreasing continually in the same proportion with the Times the other from ML to IK uniformly that both those motions begin together in M and end in K Galilaeus has demonstrated that by such motion of the point M the crooked line of a Parabola will be described Again if the point A be conceived to be moved uniformly in the straight line AB and in the same time to be carried round upon the center A by the circular motion of all the points between A and B Archimedes has demonstrated that by such motion will be described a Spiral line And seeing the circles of all these motions are concentrick in A and the interiour circle is alwayes lesse then the exteriour in the proportion of the times in which AB is passed over with uniform motion the velocity also of the circular motion of the point A will continually encrease proportionally to the times And thus far the generations of the Parabolical line MK and of the Spiral line AFGHB are like But the Uniform motion in AB concurring with circular motion in the Perimeters of all the concentrick circles describes that circle whose center is A and Perimeter BCDE and therefore that circle is by the Coroll of the first article of the 16 Chapter the aggregate of all the Velocities together taken of the point A whilst it describes the Spiral AFGHB Also the rectangle IKLM is the aggregate of all the Velocities together taken of the point M whilest it describes the crooked line MK And therefore the whole velocity by which the Parabolicall line MK is described is to the whole velocity with which the Spiral line AFGHB is described in the same time as the rectangle IKLM is to the Circle BCDE that is to the triangle AIB But because AI is bisected in K the straight lines IM AB are equal therefore the rectangle IKLM and the triangle AIB are also equal Wherefore the Spiral line AFGHB and the Parabolical line MK being described with equal velocity and in equal times are equal to one another Now in the first article of the 18 Chapter a straight line is found out equal to any Parabolical line Wherefore also a Straight line is found out equal to a given Spiral line of the first revolution described by Archimedes which was to be done 6 In the sixth Chapter which is of Method that which I should there have spoken of the Analyticks of Geometricians I thought fit to deferre because I could not there have been understood as not having then so much as named Lines Superficies Solids Equal and Unequal c. Wherefore I will in this place set down my thoughts concerning it Analysis is continual Reasoning from the Definitions of the terms of a proposition we suppose true and again from the Definitions of the terms of those Definitions and so on till we come to some things known the Composition whereof is the demonstration of the truth or falsity of the first supposition and this Composition or Demonstration is that we call Synthesis Analytica therefore is that art by which our reason proceeds from something supposed to Principles that is to prime Propositions or to such as are known by these till we have so many known Propositions as are sufficient for the demonstration of the truth or falsity of the thing supposed Synthetica is the art it self of Demonstration Synthesis therefore and Analysis differ in nothing but in proceeding forwards or backwards and Logistica comprehends both So that in the Analysis or Synthesis of any question that is to say of any Probleme the Terms of all the Propositions ought to be convertible or if they be enunciated Hypothetically the truth of the Consequent ought not onely to follow out of the truth of its Antecedent but contrarily also the truth of the Antecedent must necessarily be inferred from the truth of the Consequent For otherwise when by Resolution we are arrived at Principles we cannot by Composition return directly back to the thing sought for For those Terms which are the first in Analysis will be the last in Synthesis as for example when in Resolâ—ing we say these two Rectangles are equal and therefore their sides are reciprocally proportional we must necessarily in Compounding say the sides of these Rectangles are reciprocally proportional and therefore the Rectangles themselves are equal Which we could not say â—…ss Rectangles have their sides reciprocally proportional and Rectangles are equal were Terms convertible Now in every Analysis that which is sought is the Proportion of two quantities by which proportion a figure being described the quantity sought for may be exposed to Sense And this Exposition is the end and Solution of the question or the construction of the Probleme And seeing Analysis is reasoning from something supposed till we come to Principles that is to Definitions or to Theoremes formerly known and seeing the same reasoning tends in the last place to some Equation we can therefore make no end of Resolving till we come at last to the causes themselves of Equality and Inequality or to Theoremes formerly demonstrated from those causes and so have a sufficient number of those Theoremes for the demonstration of the thing sought for And seeing also that the end of the Analyticks is either the construction of such a Probleme as is possible or the detection of the impossibility thereof whensoever the Probleme may be solved the Analyst must not stay till he come to those things which contain the efficient cause of that whereof he is to make construction But he must of necessity stay when he comes to prime Propositions and these are Definitions These Definitions therefore must contain the efficient cause of his Construction I say of his Construction not of the Conclusion which he demonstrates for the cause of the Conclusion is contained in the premised propositions that is to say the truth of the proposition he proves is
in a straight line perpendicular to its Superficies in that point in which it is pressed Let ABCD in the first figure be a hard Body and let another Body falling upon it in the straight line EA with any inclination or without inclination press it in the point A. I say the Body so pressing not penetrating it will give to the part A an endeavour to yeild or recede in a straight Line perpendicular to the line AD. For let AB be perpendicular to AD and let BA be produced to F. If therefore AF be coincident with AE it is of it self manifest that the motion in EA will make A to endeavour in the line AB Let now EA be oblique to AD and from the point E let the straight line EC be drawn cutting AD at right angles in D and let the rectangles ABCD and ADEF be completed I have shewn in the 8th Article of the 16th Chapter that the Body will be carried from E to A by the concourse of two Uniform Motions the one in EF and its parallels the other in ED and its parallels But the motion in EF and its parallels whereof DA is one contributes nothing to the Body in A to make it endeavour or press towards B and therefore the whole endeavour which the Body hath in the inclined line EA to pass or press the Straight line AD it hath it all from the perpendicular motion or endeavour in FA. Wherefore the Body E after it is in A will have onely that perpendicular endeavour which proceeds from the motion in FA that is in AB which was to be proved 7 If a hard Body falling upon or pressing another Body penetrate the same its endeavour after its first penetration will be neither in the inclined line produced nor in the perpendicular but sometimes betwixt both sometimes without them Let EAG in the same â— figure be the inclined line produced and First let the passage through the Medium in which EA is be easier then the passage through the Medium in which AG is As soon therefore as the Body is within the Medium in which is AG it will finde greater resistance to its motion in DA and its parallels then it did whilest it was above AD and therefore below AD it will proceed with slower motion in the parallels of DA then above it Wherefore the motion which is compounded of the two motions in EF and ED will be slower below AD then above it and therefore also the Body will not proceed from A in EA produced but below it Seeing therefore the endeavour in AB is generated by the endeavour in FA if to the endeavour in FA there be added the endeavour in DA which is not all taken away by the immersion of the point A into the lower Medium the Body will not proceed from A in the perpendicular AB but beyond it namely in some straight line between AB and AG as in the line AH Secondly let the passage through the Medium EA be less easie then that through AG. The motion therefore which is made by the concourse of the motions in EF and FB is slower above AD then below it and consequently the endeavour will not proceed from A in EA produced but beyond it as in AI. Wherefore If a hard Body falling which was to be proved This Divergency of the Straight line AH from the straight line AG is that which the Writers of Opticks commonly call Refraction which when the passage is eaâ—ier in the first then in the second Medium is made by diverging from the line of Inclination towards the perpendicular and contrarily when the passage is not so easie in the first Medium by departing farther from the perpendicular 8 By the 6th Theoreme it is manifest that the force of the Movent may be so placed as that the Body moved by it may proceed in a way almost directly contrary to that of the Movent as we see in the motion of Ships For let AB in the 2d figure represent a Ship whose length from the prow to the poop is AB and let the winde lie upon it in the straight parallel lines CB DE and FG and let DE and FG be cut in E and G by a straight Line drawn from B perpendicular to AB also let BE and EG be equal and the angle ABC any angle how small soever Then between BC and BA let the straight line BI be drawn and let the Sail be conceived to be spred in the same line BI and the winde to fall upon it in the points L M and B from which points perpendicular to BI let BK MQ and LP be drawn Lastly let EN and GO be drawn perpendicular to BG and cutting BK in H and K and let HN and KO be made equal to one another and severally equal to BA I say the Ship BA by the winde falling upon it in CB DE FG and other lines parallel to them will be carried forwards almost opposite to the winde that is to say in a way almost contrary to the way of the Movent For the Winde that blowes in the Line CB will as hath been shewn in the 6th Article give to the point B an endeavour to proceed in a straight line perpendicular to the straight line BI that is in the straight line BK and to the points M and L an endeavour to proceed in the straight lines MQ and LP which are parallel to BK Let now the measure of the time be BG which is divided in the middle in E let the point B be carried to H in the time BE. In the same time therefore by the wind blowing in DM FL and as many other lines as may be drawn parallel to them the whole Ship will be applyed to the straight line HN. Also at the end of the second time EG it will be applyed to the straight line KO Wherefore the Ship will always go forwards and the angle it makes with the winde will be equal to the angle ABC how small soever that angle be and the way it makes will in every time be equal to the straight line EH I say thus it would be if the Ship might be moved with as great celerity sidewayes from BA towards KO as it may be moved forwards in the line BA But this is impossible by reason of the resistance made by the great quantity of water which presseth the side much exceeding the resistance made by the much smaller quantity which presseth the prow of the Ship so that the way the Ship makes sidewayes is scarce sensible and therefore the point B will proceed almost in the very line BA making with the winde the angle ABC how acute soever that is to say it will proceed almost in the straight line BC that is in a way almost contrary to the way of the Movent which was to be demonstrated But the Sayl in BI must
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
Crooked Lines of Parabolas and other Figures made in imitation of Parabolas 1 To find a straight Line equal to the crooked Line of a Semiparabola 2 To find a straight Line equal to the Crooked Line of the first Semiparabolaster or to the Crooked Line of any other of the Deficient Figures of the Table of the 3d. Article of the prâ—â—edent Chapter 1 AParabola being given to find a Straight Line equal to the Crooked Line of the Semiparabola Let the Parabolical Line given be ABC in the first Figure and the Diameter found be AD and the base drawn DC and the Parallelogram ADCE being completed draw the straight Line AC Then dividing AD into two equal parts in F draw FH equal and parallel to DC cutting AC in K and the parabolical line in O and between FH and FO take a mean proportional FP and draw AO AP and PC I say that the two Lines AP and PC taken together as one Line is equal to the parabolical line ABOC For the line ABOC being a parabolical line is generated by the concourse of two Motions one Uniform from A to E the other in the same time uniformly accelerated from rest in A to D. And because the motion from A to E is uniform AE may represent the times of both those motions from the beginning to the end Let therefore AE be the time and consequently the lines ordinately applyed in the Semiparabola will designe the parts of time wherein the Body that describeth the line ABOC is in every point of the same so that as at the end of the time AE or DC it is in C so at the end of the time FO it will be in O. And because the Velocity in AD is encreased uniformly that is in the same proportion with the time the same lines ordinately applyed in the Semiparabola will designe also the continual augmentations of the Impetus till it be at the greatest designed by the base DC Therefore supposing Uniform motion in the line AF in the time FK the Body in A by the concourse of the two uniform motions in AF and FK will be moved uniformly in the line AK and KO wil be the encrease of the Impetus or Swiftness gained in the time FK and the line AO will be uniformly described by the concourse of the two uniform motions in AF and FO in the time FO From O draw OL parallel to EC cutting AC in L draw LN parallel to DC cutting EC in N and the parabolical line in M and produce it on the other side to AD in I and IN IM and IL will be by the construction of a Parabola in continual proportion equal to the three lines FH FP and FO and a straight line parallel to EC passing through M will fall on P and therefore OP will be the encrease of Impetus gained in the time FO or IL. Lastly produce PM to CD in Q and QC or MN or PH will be the encrease of Impetus proportional to the time FP or IM or DQ Suppose now uniform motion from H to C in the time PH. Seeing therefore in the time FP with uniform motion and the Impetus encreased in proportion to the times is described the straight line AP and in the rest of the time and Impetus namely PH is described the line CP uniformly it followeth that the whole line APC is described with the whole Impetus and in the same time wherewith is described the parabolicall line ABC and therefore the line APC made of the two straight lines AP and PC is equal to the parabolical line ABC which was to be proved 2 To find a Straight line equal to the Crooked line of the first Semiparabolaster Let ABC be the Crooked line of the first Semiparabolaster AD the Diameter DC the Base and let the Parallelogram completed be ADCE whose Diagonal is AC Divide the Diameter into two equal parts in F and draw FH equal and parallel to DC â—utting AC in K the Crooked line in O and EC in H. Then draw OL parallel to EC cutting AC in L and draw LN parallel to the base DC cutting the Crooked line in M and the straight line EC in N and produce it on the other side to AD in I. Lastly through the point M draw PMQ parallel and equal to HC cutting FH in P and joyn CP AP and AO I say the two Straight lines AP and PC are equal to the Crooked line ABOC For the line ABOC being the Crooked line of the first Semiparabolaster is generated by the concourse of two Motions one uniform from A to E the other in the same time accelerated from rest in A to D so as that the Impetus encreaseth in proportion perpetually triplicate to that of the encrease of the time or which is all one the lengths transmitted are in proportion triplicate to that of the times of their transmission for as the Impetus or Quicknesses encrease so the Lengths transmitted encrease also And because the motion from A to E is uniform the line AE may serve to represent the time and consequently the lines ordinately drawn in the Semiparabolaster will designe the parts of time wherein the Body beginning from rest in A describeth by its motion the Crooked line ABOC And because DC which represents the greatest acquired Impetus is equal to AE the same ordinate lines will represent the several augmentations of the Impetus encreasing from rest in A. Therefore supposing uniform Motion from A to F in the time FK there will be described by the concourse of the two uniform Motions AF and FK the line AK uniformly and KO will be the encrease of Impetus in the time FK And by the concourse of the two uniform Motions in AF and FO will be described the line AO uniformly Through the point L draw the straight line LMN parallel to DC cutting the straight line AD in I the crooked line ABC in M and the straight line EC in N and through the point M the straight line PMQ parallel and equal to HC cutting DC in Q and FH in P. By the concourse therefore of the two uniform Motions in AF and FP in the time FP will be uniformly described the straight line AP and LM or OP will be the encrease of Impetus to be added for the time FO And because the proportion if IN to I L is triplicate to the proportion of I N to I M the proportion of F H to F O will also be triplicate to the proportion of F H to F P and the proportional Impetus gained in the time F P is P H. So that F H being equal to P C which designed the whole Impetus acquired by the acceleration there is no more encrease of Impetus to be computed Now in the time P H suppose an uniform motion from H to C and by the two uniform motions in C H and H P will be described
which being done the Excentricity of the Earth will be cf. Seeing therefore the annual motion of the Earth is in the Circumference of an Ellipsis of which ♑ ♋ is the greater Axis ab cannot be the lesser Axis for ab and ♑ ♋ are equal Wherefore the Earth passing through a b will either pass above ♑ as through g or passing through ♑ will fall between c and a it is no matter which Let it pass therefore through g and let gl be taken equal to the straight line ♑ ♋ and dividing gl equally in i gi will be equal to ♑ ♋ il equal to f ♋ and consequently the point i will cut the Excentricity cf into two equal parts and taking ih equal to if hi will be the whole Excentricity If now a straight line namely the line ♎ i ♈ be drawn through i parallel to the straight lines ab and ed the way of the Sunne in Summer namely the Arch ♎ g ♈ will be greater then his way in Winter by 8 degrees and ¼ Wherefore the true Aequinoxes wil be in the straight line ♎ i ♈ and therefore the Ellipsis of the Earths annual motion will not pass through a g b l but through ♎ g ♈ l. Wherfore the annual motion of the Earth is in the Ellipsis ♎ g ♈ l and cannot be the Excentricity being salved in any other line And this perhaps is the reason why Kepler against the opinion of all the Astronomers of former time thought fit to bisect the Excentricity of the Earth or according to the Ancients of the Sunne not by diminishing the quantity of the same Excentricity because the true measure of that quantity is the difference by which the Summer Arch exceeds the Winter Arch but by taking for the Center of the Ecliptick of the great Orbe the point c neerer to f so placing the whole great Orbe as much neerer to the Ecliptick of the fixed Stars towards ♋ as is the distance between c i. For seeing the whole great Orbe is but as a point in respect of the immense distance of the fixed Starres the two straight lines ♎ ♈ and ab being produced both wayes to the beginnings of Aries and Libra will fall upon the same points of the Sphere of the fixed Stars Let therefore the Diameter of the Earth mn be in the plain of the Earths annual motion If now the Earth be moved by the Sunnes simple motion in the Circumference of the Ecliptick about the Center i this Diameter will bee kept alwayes parallel to itself and to the straight line gl But seeing the Earth is moved in the Circumference of an Ellipsis without the Ecliptick the point n whilst it passeth through ♎ ♑ ♈ will go in a lesser Circumference then the point m and consequently as soon as ever it begins to be moved it will lose its parallelisme with the straight line ♑ ♋ so that mn produced will at last cut the straight line gl produced And contrarily as soon as mn is past ♈ the Earth making its way in the internal Ellipticall line ♈ l ♎ the same mn produced towards m will cut lg produced And when the Earth hath allmost finished its whole circumference the same mn shall againe make a right angle with a line drawn from the center i a little short of the point from which the Earth began its motion And there the next yeare shall be one of the Aequinoctial points namely neer the end of ♠the other shall be opposite to it neer the end of ♓ And thus the points in which the Days and Nights are made equall doe every year fall back but with so slow a motion that in a whole year it makes but 51 first minutes And this relapse being contrary to the order of the Signes is commonly called the Praecession of the Aequinoxes Of which I have from my former Suppositions deduced a possible cause which was to be done According to what I have said concerning the cause of the Excentricity of the Earth and according to Kepler who for the cause thereof supposeth one part of the Earth to be affected to the Sunne the other part to be disaffected the Apogaeum Perigaeum of the Sunne should be moved every year in the same order and with the same velocity with which the Aequinoctiall points are moved and their distance from them should allwayes be the quadrant of a circle which seems to be otherwise For Astronomers say that the Aequinoxes are now the one about 28 degrees gone back from the first Star of Aries the other as much from the beginning of Libra So that the Apogaeum of the Sunne or the Aphelium of the Earth ought to be about the 28th degree of Cancer but it is reckoned to be in the 7th degree Seeing therefore we have not sufficient evidence of the 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 that so it is it is in vaine to seek for the 〈◊〉 〈◊〉 〈◊〉 〈◊〉 〈◊〉 why it is so Wherefore as long as the motion of the Apogaeum is not observable by reason of the slownesse thereof and as long as it remaiues doubtful whether their distance from the Aequinoctiall points be more or lesse then a quadrant precisely so long it may be lawfull for me to thinke they proceed both of them with equall velocity Also I doe not at all meddle with the causes of the Excentricities of Saturne Jupiter Mars and Mercury Neverthelesse seeing the Excentricity of the Earth may as I have shewne be caused by the unlike constitution of the several parts of the Earth which are alternately turned towards the Sunne it is credible also that like effects may be produced in these other Planets from their having their Superficies of unlike parts And this is all I shall say concerning Sidereal Philosophy And though the causes I have here supposed be not the true causes of these Phaenomena yet I have demonstrated that they are sufficient to produce them according to what I at first propounded CHAP. XXVII Of Light Heat and of Colours 1 Of the immense Magnitude of some Bodies and the unspeakable Littleness of others 2 Of the cause of the Light of the Sun 3 How Light heateth 4 The generation of Fire from the Sunne 5 The generation of Fire from Collision 6 The cause of Light in Glow-wormes Rotten Wood and the Bolonian Stone 7 The cause of Light in the concussion of Sea-water 8 The cause of Flame Sparks and Colliquation 9 The cause why wet Hay sometimes burns of its own accord Also the cause of Lightning 10 The cause of the force of Gunpowder and what is to be ascribed to the Coals what to the Brimstone and what to the Nitre 11 How Heat is caused by Attrition 12 The distinction of Light into First Second c. 13 The causes of the Colours we see in looking through a Prisma of Glass namely of Red Yellow Blue and Violet Colour 14 Why the Moon and the Starres appear redder in the Horizon then in
to be setled any where as at H. If now the heat of the aire be augmented the water will descend below H and if the heat be diminished it will ascend above it Which though it be certainely known to be true by experience the cause neverthelesse hath not as yet been discovered In the 6 and 7 articles of the 27th chapter where I consider the cause of Cold I have shewne that fluid Bodies are made colder by the pressure of the aire that is to say by a constant Wind that presseth them For the same cause it is that the Superficies of the water is pressed at F and having no place to which it may retire from this pressure besides the cavity of the Cylinder between H and E it is therefore necessarily forced thither by the Cold and consequently it ascendeth more or lesse according as the Cold is more or lesse encreased And againe as the Heat is more intense or the Cold more remisse the same water will be depressed more or lesse by its own Gravity that is to say by the cause of Gravity above explicated 13 Also Living creatures though they be Heavy can by Leaping Swimming Flying raise themselvs to a certain degree of height But they cannot do this except they be supported by some resisting Body as the Earth the Water and the Aire For these motions have their beginning from the contraction by the helpe of the Muscles of the Body animate For to this contraction there succeedeth a distension of their whole Bodies by which distension the Earth the Water or the Aire which supporteth them is pressed and from hence by the reaction of those pressed Bodies Living Creatures acquire an endeavour upwards but such as by reason of the Gravity of their Bodies is presently lost againe By this endeavour therefore it is that Living creatures rayse themselues up a little way by Leaping but to no great purpose but by Swimming Flying they raise themselves to a greater height because before the effect of their endeavour is quite extinguished by the Gravity of their bodies they can renew the same endeavour againe That by the power of the Soule without any antecedent contraction of the Muscles or the helpe of something to support him any man can be able to raise his Body upwards is a childish conceipt For if it were true a man might raise himselfe to what height he pleased 14 The diaphanous Medium which surrounds the Eie on all fides is invisible Nor is Aire to be seen in Aire nor Water in Water nor any thing but that which is more opacous But in the confines of two diaphanous Bodies one of them may be distinguished from the other It is not therefore a thing so very ridiculous for ordinary people to think all that Space empty in which we say is Aire it being the worke of Reason to make us conceive that the Aire is any thing For by which of our Senses is it that we take notice of the Aire seeing we neither See nor Hear nor Tast nor Smell nor Feel it to be any thing When we feel Heat we do not impute it to the Air but to the Fire nor do we say the aire is Cold but we our selves are Cold and when we feel the Wind we rather think something is comming then that any thing is already come Also we do not at al feel the waight of water in water much less of air in air That we come to know that to be a Body which we call Aire it is by Reasoning but it is from one Reason onely namely because it is impossible for remote Bodies to work upon our Organs of Sense but by the help of Bodies intermediate without which we could have no sense of them till they came to be contiguous Wherefore from the Senses alone without reasoning from effects we cannot have sufficient evidence of the nature of Bodies For there is under-ground in some Mines of Coles a certain matter of a middle nature between Water and Aire which nevertheless cannot by Sense be distinguished from aire for it is as Diaphanous as the purest aire and as farre as Sense can judge equally penetrable But if we look upon the effect it is like that of water For when that matter breaks out of the Earth into one of those Pits it fils the same either totally or to some degree and if a Man or Fire be then let down into it it extinguishes them in almost as little time as water would do But for the better understanding of this Phaenomenon I shall describe the 6th figure In which let A B represent the pit of the Mine and let part thereof namely C B be supposed to be filled with that matter If now a lighted Cādle be let down into it below C it wil as suddenly be extinguished as if it were thrust into water Also if a grate filled with coles throughly kindled and burning never so brightly be let down as soon as ever it is below C the fire will begin to grow pale and shortly after losing its light be extinguished no otherwise then if it were quenched in water But if the grate be drawn up again presently whilest the coles are still very hot the fire will by little and little be kindled again and shine as before There is indeed between this matter water this considerable difference that it neither wetteth nor sticketh to such things as are put down into it as water doth which by the moisture it leaveth hindereth the kindling again of the matter once extinguished In like manner if a Man be let down below C he will presently fall into a great difficulty of breathing and immediately after into a swoun and die unless he be suddenly drawn up again They therefore that go down into these pits have this custome that as soon as ever they feel themselves sick they shake the rope by which they were let down to signifie they are not well and to the end that they may speedily be pulled up again For if a man be drawn out too late void of sense and motion they digg up a Turff and put his face and mouth into the fresh earth by which means unless he be quite dead he comes to himself again by little and little and recovers life by the breathing out as it were of that suffocating matter which he had sucked in whilest he was in the pit almost in the same manner as they that are drowned come to themselves again by vomiting up the water But this doth not happen in all Mines but in some onely and in those not alwayes but often In such Pits as are subject to it they use this remedy They dig another pit as DE close by it of equal depth and joyning them both together with one common channel EF they make a Fire in the bottom E which carries out at D the aire contained in the pit DE and this draws with it the aire contained in the channel EF which