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
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
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
two Times betwixt which there is no other Time are called IMMEDIATE A B C as AB BC. And any two Spaces as well as Times are said to be CONTINUALL when they have one common part A B C D as AC BD where the part BC is common and more Spaces and Times are Continual when every two which are next one another are Continual 11 That Part which is between two other Parts is called a MEAN that which is not between two other parts an EXTREME And of Extremes that which is first reckoned is the BEGINNING and that which last the END and all the Means together taken are the WAY Also Extreme Parts and Limits are the same thing And from hence it is manifest that Beginning and End depend upon the order in which we number them and that to Terminate or Limit Space and Time is the same thing with imagining their Beginning and End as also that every thing is FINITE or INFINITE acording as we imagine or not imagine it Limited or Terminated every way and that the Limits of any Number are Unities and of these that which is the first in our Numbering is the Beginning and that which we number last is the End When we say Number is Infinite we mean only that no Number is expressed for when we speak of the Numbers Two Three a Thousand c. they are always Finite But when no more is said but this Number is Infinite it is to be understood as if it were said this Name Number is an Indefinite Name 12 Space or Time is said to be Finite in Power or Terminable when there may be assigned a Number of finite Spaces or Times as of Paces or Hours than which there can be no greater Number of the same measure in that Space or Time and Infinite in Power is that Space or Time in which a greater Number of the said Paces or Hours may be assigned than any Number that can be given But we must note that although in that Space or Time which is Infinite in Power there may be numbered more Paces or Hours then any number that can be assigned yet their number will alwayes be Finite for every Number is Finite And therefore his Ratiocination was not good that undertaking to prove the World to be Finite reasoned thus If the world be Infinite then there may be taken in it some Part which is distant from us an Infinite number of Paces But no such Part can be taken wherefore the world is not infinite because that Consequence of the Major Proposition is false for in an Infinite space whatsoever we take or design in our Mind the distance of the same from us is a Finite space for in the very designing of the place thereof we put an End to that space of whch we our selves are the Beginning and whatsoever any man with his Mind cuts off both wayes from Infinite he determines the same that is he makes it Finite Of Infinite Space or Time it cannot be said that it is a Whole or One not a Whole because not compounded of Parts for seeing Parts how many soever they be are severally Finite they will also when they are all put together make a whole Finite Nor One because nothing can be said to be One except there be Another to compare it with but it cannot be conceived that there are two Spaces or two Times Infinite Lastly when we make question whether the World be Finite or Infinite we have nothing in our Minde answering to the Name World for whatsoever we Imagine is therefore Finite though our Computation reach the fixed Stars or the ninth or tenth nay the thousanth Sphere The meaning of the Question is this onely whether God has actually made so great an Addition of Body to Body as we are able to make of Space to Space 13 And therefore that which is commonly said that Space and Time may be divided Infinitely is not to be so understood as if there might be any Infinite or Eternal Division but rather to be taken in this sense Whatsover is Divided is divided into such Parts as may again be Divided or thus The Least Divisible thing is not to be given or as Geometricians have it No Quantity is so small but a Less may be taken which may easily be demonstrated in this manner Let any Space or Time that which was thought to be the Least Divisible be divided into two equal Parts A and B. I say either of them as A may be divided again For suppose the Part A to be contiguous to the Part B of one side and of the other side to some other Space equal to B. This whole Space therefore being greater then the Space given is divisible Wherefore if it be divided into two equal Parts the Part in the middle which is A will be also divided into two equal Parts and therefore A was Divisible CHAP. VIII Of Body and Accident 1 Body defined 2 Accident defined 3 How an Accident may be understood to be in its subject 4 Magnitude what it is 5 Place what it is and that it is Immoveable 6 What is Full and Empty 7 Here There Somewhere what they signifie 8 Many Bodies cannot be in One place nor One Body in Many places 9 Contiguous and Continual what they are 10 The definition of Motion No Motion intelligible but with Time 11 What it is to be at Rest to have been Moved and to be Moved No Motion to be conceived without the conception of Past and Future 12 A Point a Line Superficies and Solid what they are 13 Equal Greater and Lesse in Bodies and Magnitudes what they are 14 One and the same Body has alwayes one and the the same Magnitude 15 Velocity what it is 16 Equal Greater and Lesse in Times what they are 17 Equal Greater and Lesse in Velocity what 18 Equal Greater and Lesse in Motion what 19 That which is at Rest will alwayes be at Rest except it be Moved by some external thing and that which is Moved will alwayes be Moved unless it be hindered by some external thing 20 Accidents are Generated and Destroyed but Bodies not so 21 An Accident cannot depart from its Subject 22 Nor be Moved 23 Essence Form and Matter what they are 24 First Matter what 25 That the whole is greater then any Part thereof why demonstrated 1 HAving understood what Imaginary Space is in which we supposed nothing remaining without us but all those things to be destroyed that by existing heretofore left Images of themselves in our Minds let us now suppose some one of those things to be placed again in the World or created anew It is necessary therefore that this new created or replaced thing do not onely fill some part of the Space above-mentioned or be coincident and coextended with it but also that it have no dependance upon our thought And this is that which for the Extension of it we commonly call Body and because it depends
that the Proportion of the first Antecedent to the first Consequent is the same with that of the second Antecedent to the second Consequent And when four Magnitudes are thus to one another in Geometrical Proportion they are called Proportionals and by some more briefly Analogisme And Greater Proportion is the Proportion of a Greater Antecedent to the same Consequent or of the same Antecâ—dent to a Less Consequent and when the Proportion of the first Antecedent to the first Consequent is greater then that of the second Antecedent to the second Consequent the four Magnitudes which are so to one another may be called Hyperlogisme Less Proportion is the Proportion of a Less Antecedent to the same Consequent or of the same Antecedent to a Greater Consequent and when the Proportion of the first Antecedent to the first Consequent is less then that of the second to the second the four Magnitudes may be called Hypologisme 5 One Arithmetical Proportion is the Same with another Arithmetical Proportion when one of the Antecedents exceeds its Consequent or is exceeded by it as much as the other Antecedent exceeds its Consequent or is exceeded by it And therefore in four Magnitudes Arithmetically Proportional the sum of the Extremes is equal to the sum of the Means For if A. B C. D be Arithmetically Proportional and the Difference on both sides be the same Excess or the same Defect E then B+C if A be greater then B will be equal to A − E+C and A+D will be equal to A+C − E But A − E+C and A+C − E are equal Or if A be less then B then B+C will be equal to A+E+C and A+D will be equal to A+C+E But A+E+C and A+C+E are equal Also if there be never so many Magnitudes Arithmetically Proportional the Sum of them all will be equal to the Product of half the number of the Terms multiplyed by the Sum of the Extremes For if A. B C. D E. F be Arithmetically Proportional the Couples A+F B+E C+D will be equal to one another and their Sum will be equal to A+F multiplyed by the number of their Combinations that is by half the number of the Terms If of four Unequal Magnitudes any two together taken be equal to the other two together taken then the greatest and the least of them will be in the same Combination Let the Unequal Magnitudes be A B C D and let A+B be equal to C+D let A be the greatest of them all I say B will be the least For if it may be let any of the rest as D be the least Seeing therefore A is greater then C and B then D A+B will be greater then C+D which is contrary to what was supposed If there be any four Magnitudes the Sum of the greatest and least the Sum of the Means the difference of the two greatest and the difference of the two least will be Arithmetically Proportional For let there be four Magnitudes whereof A is the greatest D the least and B and C the Means I say A+D B+C A − B. C − D are Arithmetically Proportional For the difference between the first Antecedent and its Consequent is this A+D − B − C and the difference between the second Antecedent and its Consequent this A − B − C+D but these two Differences are equal and therefore by this 5th Article A+D B+C A − B. C − D are Arithmetically Proportional If of four Magnitudes two be equal to the other two they will be in reciprocal Arithmetical Proportion For let A+B be equal to C+D I say A. C D. B are Arithmetically Proportional For if they be not let A. C D. E supposing E to be greater or less then B be Arithmetically Proportional and then A+E will be equal to C+D wherefore A+B and C+D are not equal which is contrary to what was supposed 6 One Geometrical Proportion is the same with another Geometrical Proportion when the same Cause producing equal Effects in equal Times determines both the Proportions If a Point Uniformly moved describe two Lines either with the same or different Velocity all the parts of them which are contemporary that is which are described in the same time will be Two to Two in Geometrical Proportion whether the Antecedents be taken in the same Line or not For from the point A in the 10 Figure at the end of the 14 Chapter let the two Lines A D A G be described with Uniform Motion and let there be taken in them two parts AB AE and again two other parts AC AF in such manner that AB AE be contemporary and likewise AC AF contemporary I say first taking the Antecedents AB AC in the Line AD and the Consequents AE AF in the Line AG that AB AC AE AF are Proportionals For seeing by the 8th Chapter and the 15 Article Velocity is Motion considered as determined by a certain Length or Line in a certain Time transmitted by it the quantity of the Line AB will be determined by the Velocity and Time by which the same AB is described and for the same reason the quantity of the Line AC will be determined by the Velocity and Time by which the same AC is described and therefore the proportion of AB to AC whether it be Proportion of Equality or of Excess or Defect is determined by the Velocities and Times by which AB AC are described But seeing the Motion of the Point A upon AB and AC is Uniform they are both desribed with equal Velocity and therefore whether one of them have to the other the Proportion of Majority or of Minority the sole cause of that Proportion is the difference of their Times and by the same reason it is evident that the proportion of AE to AF is determined by the difference of their Times onely Seeing therefore AB AE as also AC AF are contemporary the difference of the Times in which AB and AC are described is the same with that in which AE and AF are described Wherfore the proportion of AB to AC and the proportion of AE to AF are both determined by the same Cause But the Cause which so determines the proportion of both works equally in equal Times for it is Uniform Motion and therefore by the last precedent Definition the proportion of AB to AC is the same with that of AE to AF and consequently AB AC AF. AF are Proportionals which is the first Secondly taking the Antecedents in different Lines I say AB AE AC AF are Proportionals For seeing AB AE are described in the same Time the difference of the Velocities in which they are described are the sole Cause of the proportion they have to one another And the same may be said of the proportion of AC to AF. But seeing both the Lines AD and AG are passed over by Uniform Motion the difference of the Velocities in which AB AE are described will be the same with the
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
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
to be blamed who made use of the Quadratrix for the finding out of a straight line equal to the arch of a Circle and Pappus himself was he faulty when he found out the trisection of an Angle by the help of an Hyperbole Or am I in the wrong who think I have found out the construction of both these Problemes by the Rule and Compass onely Neither they nor I. For the Ancients made use of this Analysis which proceeds by the Powers and with them it was a fault to do that by a more remote Power which might be done by a neerer as being an argument that they did not sufficiently understand the nature of the thing The virtue of this kind of Analysis consists in the changing and turning and tossing of Rectangles and Analogismes and the skill of Analysts is meer Logick by which they are able methodically to find out whatsoever lies hid either in the Subject or Predicate of the ConclusioÌ„ sought for But this doth not properly belong to Algebra or the Analyticks Specious Symbolical or Cossick which are as I may say the Brachygraphy of the Analyticks and an art neither of teaching nor learning Geometry but of registring with brevity and celerity the inventions of Geometricians For though it be easie to discourse by Symbols of very remote propositions yet whether such discourse deserve to be thought very profitable when it is made without any Ideas of the things themselves I know not CHAP. XXI Of Circular Motion 1 In Simple Motion every Straight Line taken in the Body moved is so carried that it is always parallel to the places in which it formerly was 2 If Circular Motion be made about a resting Center and in that Circle there be an Epicyle whose revolution is made the contrary way in such manner that in equal times it make equal angles every Straight Line taken in that Epicycle will be so carried that it will alwayes be parallel to the places in which it formerly was 3 The properties of Simple Motion 4 If a fluid Body be moved with simple Circular Motion all the points taken in it will describe their Circles in times proportional to the distances from the Center 5 Simple Motion dissipates Heterogeneous and congregates Homogeneous Bodies 6 If a Circle made by a Movent moved with Simple Motion be commensurable to another Circle made by a point which is carried about by the same Movent all the points of both the Circles will at some time return to the same situation 7 If a Sphere have Simple Motion its Motion will more dissipate Heterogeneous Bodies by how much it is more remote from the Poles 8 If the Simple Circular Motion of a fluid Body be hindered by a Body which is not fluid the fluid Body will spread it self upon the Superficies of that Body 9 Circular Motion about a fixed Center casteth off by the Tangent such things as lie upon the Circumference and stick not to it 10 Such things as are moved with Simple Circular Motition beget Simple Circular Motion 11 If that which is so moved have one side hard and the other side fluid its Motion will not be perfectly Circular 1 I Have already defined Simple Motion to be that in which the several points taken in a moved Body do in several equal times describe several equal arches And therefore in Simple Circular Motion it is necessary that every Straight Line taken in the Moved Body be alwayes carried parallel to itself which I thus demonstrate First let A B in the first figure be any Straight Line taken in any Solid Body and let AD be any arch drawn upon any Center C and Radius CA. Let the point B be understood to describe towards the same parts the arch BE like and equall to the arch AD. Now in the same time in which the point A transmits the arch AD the point B which by reason of its simple motion is supposed to be carried with velocity equall to that of A will transmit the arch BE and at the end of the same time the whole AB will be in DE and therefore AB and DE are equall And seeing the arches AD and BE are like and equall their subtending straight lines AD and BE will also be equall and therefore the four sided figure ABDE will be a parallelogram Wherefore AB is carried parallel to it selfe And the same may be proved by the same method if any other straight line be taken in the same moved Body in which the straight line AB was taken So that all straight lines taken in a Body moved with Simple Ciâ—cular Motion will be carried parallel to themselves Coroll 1 It is manifest that the same will also happen in any Body which hath Simple Motion though not Circular For all the points of any straight line whatsoever will describe lines though not Circular yet equall so that though the crooked lines AD and BE were not arches of Circles but of Parabolas Ellipses or of any other figures yet both they and their Subtenses and the straight lines which joyne them would be equal and parallel Coroll 2 It is also manifest that the Radii of the equall circles AD and BE or the Axis of a Sphere will be so carried as to be allwayes parallel to the places in which they formerly were For the straight line BF drawn to the center of the arch BE being equall to the Radius AC will also be equall to the straight line FE or CD and the angle BFE will be equall to the angle ACD Now the intersection of the straight lines CA and BE being at G the angle CGE seeing BE and AD are parallel will be equal to the angle DAC But the angle EBF is equal to the same angle DAC and therefore the angles CGE and EBF are also equal Wherefore AC and BF are parallel which was to be demonstrated 2 Let there be a Circle given in the second figure whose center is A and Radius AB and upon the center B and any Radius BC let the Epicycle CDE be described Let the center B be understood to be carried about the center A and the whole Epicycle with it till it be coincident with the Circle FGH whose center is I and let BAI be any angle given But in the time that the center B is moved to I let the Epicycle CDE have a contrary revolution upon its own center namely from E by D to C according to the same proportions that is in such manner that in both the Circles equal angles be made in equal times I say EC the Axis of the Epicycle will be alwayes carried parallel to it self Let the angle FIG be made equal to the angle BAI IF and AB will therefore be parallel and how much the Axis AG has departed from its former place AC the measure of which progression is the angle CAG or CBD which I suppose equal to it so much in the same time has the
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
that is stricken be not onely sufficiently hard but have also the particles of which it consisteth like to one another both in hardness and figure such as are the particles of Glass and Metals which being first melted do afterwards settle and harden the Sound it yeildeth will because the motions of its parts and their reciprocations are like and Uniform be Uniform and pleasant and be more or less Lasting according as the Body stricken hath gteater or less magnitude The possible cause therefore of Sounds Uniform and Harsh and of their longer or shorter Duration may be one and the same likeness and unlikeness of the internal parts of the Sounding Body in respect both of their figure and hardness Besides if two plain Bodies of the same matter and of equal thickness do both yeild an Uniform Sound the Sound of that Body which hath the greatest extent of length will be the longest heard For the motion which in both of them hath its beginning from the point of percussion is to be propagated in the greater Body through a greater Space and consequently that propagation will require more time and therefore also the parts which are moved will require more time for their return Wherefore all the reciprocations cannot be finished but in longer time and being carried to the Eare will make the Sound last the longer And from hence it is manifest that of hard Bodies which yeild an Uniform Sound the Sound lasteth longer which comes from those that are round and hollow then from those that are plain if they be like in all other respects For in circular lines the action which begins at any point hath not frō the figure any end of its propagation because the line in which it is propagated returns again to its beginning so that the figure hinders not but that the motion may have infinite progression whereas in a plain every line hath its magnitude finite beyond which the action cannot proceed If therefore the matter be the same the motion of the parts of that Body whose figure is round and hollow wil last longer then of that which is plain Also if a string which is stretched be fastned at both ends to a hollow Body and be stricken the Sound will last longer then if it were not so fastned because the trembling or reciprocation which it receives from the stroke is by reason of the connexion communicated to the hollow Body and this trembling if the hollow Body be great will last the longer by reason of that greatness Wherefore also for the reason above mentioned the Sound will last the longer 9 In Hearing it happens otherwise then in Seeing that the action of the Medium is made stronger by the Wind when it blows the same way and weaker when it blows the contrary way The cause whereof cannot proceed from any thing but the different generation of Sound and Light For in the generation of Light none of the parts of the Medium between the object and the Eie are moved from their own places to other places sensibly distant but the action is propagated in spaces imperceptible so that no contrary Wind can diminish nor favourable Winde encrease the Light unless it be so strong as to remove the Object further off or bring it nearer to the Eie For the Wind that is to say the aire moved doth not by its interposition between the object and the eie worke otherwise then it would doe if it were stil and calme For where the pressure is perpetuall one part of the aire is no sooner carried away but another by succeeding it receives the same impression which the part carried away had received before But in the generation of Sound the first collision or breaking asunder beateth away driveth out of its place the nearest part of the aire and that to a considerable distance and with considerable velocity and as the circles grow by their remotenesse wider and wider so the aire being more more dissipated hath also its motion more more weakned Whensoever therfore the air is so stricken as to cause Sound if the Wind fall upon it it will move it all neerer to the Eare if it blow that way and further from it if it blow the contrary way so that according as it blowes from or towards the Object so the Sound which is heard will seeme to come from a neerer or remoter place and the action by reason of the unequall distances be strengthened or debilitated From hence may be understood the reason why the voice of such as are said to speake in their bellies though it be uttered neer hand is neverthelesse heard by those that suspect nothing as if it were a great way off For having no former thought of any determined place from which the voice should proceed and judging according to the greatesse if it be weake they thinke it a great way off if strong neer These Ventriloqui therefore by forming their voice not as others by the emission of their breath but by drawing it inwards doe make the same appear small and weake which weaknesse of the voice deceives those that neither suspect the artifice nor observe the endeavour which they use in speaking and so instead of thinking it weake they thinke it farre off 10 As for the Medium which conveighs Sound it is not Aire onely For Water or any other Body how hard soever may be that Medium For the Motion may be propagated perpetually in any hard continuous Body but by reason of the difficulty with which the parts of hard Bodies are moved the motion in going out of hard matter makes but a weak impression upon the Aire Nevertheless if one end of a very long and hard beam be stricken the eare be applyed at the same time to the other end so that when the action goeth out of the beam the aire which it striketh may immediately be received by the eare and be carried to the Tympanum the Sound will be considerably strong In like manner if in the night when all other noyse which may hinder Sound ceaseth a man lay his eare to the ground he will hear the Sound of the steps of Passengers though at a great distance because the motion which by their treading they communicate to the earth is propagated to the eare by the uppermost parts of the earth which receiveth it from their feet 11 I have shewn above that the difference between Grave and Acute Sounds consisteth in this that by how much the shorter the time is in which the reciprocations of the parts of a Body stricken are made by so much the more Acute will be the Sound Now by how much a Body of the same bigness is either more heavy or less stretched by so much the longer will the reciprocations last and therefore heavier and less stretched Bodies if they be like in all other respects will yeild a Graver Sound then such as be lighter and more stretched 12 For the finding out