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 M E C H A N I C Sj 
 
 O R, T H E ^ . . ^ 
 
 D O C T R I N WS^''^\ 
 
 O F -,4 '-S- ,i^' 
 
 M O T I O N?^ 
 
 COMPREHENDING, 
 
 I. The general Laws of Motion. 
 
 II. The Descent of Bodies perpendicularly, and 
 
 down inclined Planes, and alfo in Curvk 
 Surfaces. The Motion of Pendulums. 
 
 III. Centers of Gravity. The Equilibrium of 
 Beams of Timber, and their Forces and Di- 
 rections. 
 
 IV. The Mechanical Powers. 
 
 V. The comparative Strength of Timber, and 
 
 its Stress. The Powers of Engines, their 
 Motion, and Friction. 
 
 VI. Hydrostatics and Pneumatics. 
 
 Da veniam fcriptis^ quorum non gloria nobis 
 caufa, fed utilitas officiumque^ fuit. 
 
 Ovid de Ponto III. 
 
 LONDON: 
 
 Printed for J. N o u r s e, in the Strand j 
 
 Bookfeller in Ordinary to his MAJESTY. 
 
 MDCCLXIX.
 
 stack 
 Annex 
 
 THE 5^ 
 
 PREFACE. 
 
 \\ 
 
 , TLTA V in G fome years fince wriiten a large hook 
 of Mechanics^ which is fold by Mejp^' Robinlbn, 
 and Co. in Pater-nofter-Row. / have here given a 
 Jhort ahjtraSl of that book, as it falls properly into 
 this cotirfe •, and efpecially as that branch of fcience, 
 is of fuch extenfroe ufe in the affairs of human life. 
 1 do not in the leaf, defign this to interfere with the 
 other book, being rather an introduction to it, as it 
 explains feveral things in it more at large ; particu- 
 larly in the firfi feClion, as being of univerfal extent 
 and life \ and likewife in fever al other parts of the 
 book, efpecially fuch as have been obje^ed to by igno- 
 r.ant writers. I have alfo added feveral things not 
 mentioned in the other book, which are more fimple 
 and eafy, and more proper for learners. So that this 
 (hort treatife may be looked upon as an introdu^ion to 
 the other book, and will doubtlefs facilitate the read- 
 ing of it. As to the higher and more difficult matters^ 
 4ii few care to trouble their heads about them, I have 
 faid little of them here, being not fo proper for an in- 
 troduction, ^0 mention one or two things ; / had 
 taken a great deal of pains to find out the true form 
 of a bridge, that fhall be the firongefl, and of a 
 JJjip that fhall fail the fo.flefl -, both upon principles 
 that I know to be as certain and demoyiftrative as the 
 Elements of Euclid -, both thefe you have in the other 
 book. But^ as we have no occqfon in England, for 
 
 A 2 the 
 
 -i fi'1 "iri 95
 
 J DEFINITIONS. 
 
 lute motion. But when compared with others in 
 motion, it is called relative motion- 
 
 y. Dire^ion of motion is the courfe or way the 
 body tends, or the line it moves in. 
 
 8 . ^antity of motion, is the motion a body has, 
 confidered both in regard to its velocity and quan- 
 tity of matter. This is alfo called the Momentum 
 of a body. 
 
 9 Fis inertia, is the innate force of matter, by 
 which it refills any change, driving to preferve its 
 prefent ftate of reft or motion. 
 
 10. Grav ty is that force wherewith a body en- 
 deavours to fall downwards. It is called abfolute 
 gravity in empty fpace •, and relative gravity when 
 immerfed in a fluid. 
 
 1 1 . Specific gravity, is the greater or lefTer weight 
 of bodies of the fame magnitude, or the proportion 
 between their weights. This proceeds from the na- 
 tural denfity of bodies 
 
 12. Center of gravity^ is a certain point of a 
 body, upon which, the body when fufpended, 
 will reft in any pofition. 
 
 13. Center of motion, is a fixed point about 
 which a body moves. And the axis of motion is a 
 fixed line it moves about. 
 
 14. Po'wer and weight, when oppofed to one 
 another, fignify the body that moves another, and 
 the other which is moved. The body which be- 
 gins and communicates motion is the power j and 
 that which receives the motion, is the weight. 
 
 15. Equilihium is the balance of two or more 
 forces, fo as to remain at reft. 
 
 16. Machine or Engine, is any inftrument to 
 move bodies, made of levers, wheels, pullies, &c, 
 
 17. Mechanic po-i .s, are the ballance, lever, 
 wheel, pulley, fcrew und wedge. 
 
 18. Strefs is the effefl any force has to break 
 a beam, or any other body j and Jtrength is the 
 
 refiftance
 
 P O S T U L A T A. ^ 
 
 refinance it is able to make againft any flraining 
 force. 
 
 19. Frisian is the refiftance which a machine 
 fuffers, by the parts rubbing againft one another. 
 
 POSTULATA. 
 
 1 . That a fmall part of the furface of the earth 
 may be looked upon as a plane. For tho' the earth 
 be round, yet fuch a fmall part of it as we have 
 any occafion to confider, does not fenfibly differ 
 from a plane. 
 
 2. That heavy bodies defcend in lines parallel to 
 one another. For tho' they all tend to a point 
 which is the center of the earth, yet that center is 
 at fuch a diftance that thefe lines differ infenfibly 
 from parallel lines. 
 
 3. The fame body is of the fame weight in all 
 places on or near the earth's furface. For the dif- 
 ference is not fenfible in the feveral places we can 
 goto. 
 
 4. Tho' all matter is rough, and all engines im- 
 perfeft •, yet for the eafe of calculation, we muft 
 fuppofe all planes perfe6tly even -, all bodies 
 perfeftly fmooth -, and all bodies and machines to 
 move without friftion or refiftance ; all lines ftreight 
 and inflexible, without weight or thicknefs j cords 
 extremely pliable, and fo on. 
 
 AXIOMS. 
 
 1. Every body endeavours to remain in its pre- 
 fent ftate, whether it be at reft, or moving uni- 
 formly in a right line. 
 
 2. The alteration of motion by any external 
 force is always proportional to that force, and in 
 diredion of the right line in which the force ads. 
 
 3. Action and re-a6lion, between any two bodies, 
 are equal and contrary. 
 
 B 2 4. The
 
 4 AXIOMS. 
 
 4. The motion of airt body is made up of the 
 fum of the motions of all the parts. 
 
 5. The weights of all bodies in the fame place, 
 are proportional to the quantities of matter they 
 contain, without any regard to their figure. 
 
 6. The vis inertiae of any body, is proportional 
 to the quantity of matter. 
 
 7. Every body will defcend to the loweft place 
 
 It can get to. 
 
 8. Whatever fuftains a heavy body, bears all 
 the weight of it. 
 
 9. Two equal forces afting againft one another 
 in contrary dire6tions -, deftroy one anothers ef- 
 fe<5ts. And unequal forces a6l only with the dif- 
 ference of them. 
 
 10. When a body is kept in equilibrio ; the con- 
 trary forces in any line of direftion are equal. 
 
 11. If a certain force generate any motion; an 
 equal force afting in a contrary direction, will de- 
 ftroy as much motion in the fame time. 
 
 12. If a body be acSted on by any power in a 
 given diretftion. It is all one in what point of that 
 line of direction, the power is applied. 
 
 ic^. If a body is drawn by a rope, all the parts 
 of the rope are equally ftretched. And the force 
 in any part ads in direftion of that part. And it 
 is the fame thing whether the rope is drawn out 
 at length, or goes over feveral pullics. 
 
 14. If feveral forces at one end of a lever, afl 
 againft feveral forces at the other end -, the lever 
 acts and is a<5ted on in diredftion of its length. 
 
 SEC T.
 
 f 5 ] 
 
 SECT. I. 
 
 "The General Laws of MoTioisr. 
 
 PROP. I. 
 
 ^H E quantities of matter in all bodies^ are in the 
 compound ratio of their magnitudes and denjities. 
 
 For (Def. 3.) in bodies of the fame magnitudes, 
 the quantities of matter will be as the denfities. 
 Increafe the magnitude in any ratio, and the quan- 
 tity of matter is increafed in the fame ratio. Con- 
 fequently the quantity of matter is in the com- 
 pound ratio of the denfity and magnitude. 
 
 Cor. I . In two fimilar bodies^ the quantities of 
 matter are as the denfities^ and cubes of the diameters. 
 
 For the magnitudes of bodies are as the cubes 
 of the diameters. 
 
 Cor. 2. The quantities of matter are as the mag- 
 nitudes and fpecifc gravities. 
 
 For (by Def. 3. and 11.) the denfities of bodies 
 are as their fpecific gravities. 
 
 PROP. II. 
 
 The quantities of motion in all movijig bodies, are 
 in the compound ratio of the quantities of matter and 
 the velocities. 
 
 For if the velocities be equal, the quantities of 
 motion will manifeftly be as the quantities of mat- 
 ter. Increafe the velocity in any ratio, and the 
 c^uantity of motion will be increafed in the fame 
 
 B 3 ratio.
 
 6 GENERAL LAWS 
 
 ratio. Therefore it follows univerfally, that the 
 quantities of motion are in the compound ratio of 
 the velocities and quantities of matter. 
 
 Cor. Hence if the body he the fame^ the motion is 
 /IS the velocity. And if the velocity be the fame, the 
 motion is as the body or quantity of matter. 
 
 PROP. in. 
 
 In all bodies moving uniformly, the fpaces defcrihed, 
 are in the compound ratio cf the velocities and the times 
 of their defcription. 
 
 For in any moving body, the greater the veloci- 
 ty^ the greater is the fpace defcribed ; that is, the 
 fpace will be as the velocity. And in twice or 
 thrice the time, &c. the fpace will be twice or 
 thrice as great ; that is, the fpace will be increafed 
 in proportion to the time. Therefore univerfally 
 the fpace is in the compound ratio of the velocity, 
 and the time of defcription. 
 
 Cor. I. The time of defcribing any fpace, is as the 
 fpace directly and velocity reciprocally ; or as the fpace 
 divided by the velocity. And if the velocity be the 
 fame, the time is as the fpace. And if the fpace be 
 the fame, the time is reciprocally as the velocity. 
 
 Cor. 2. The velocity of a moving body, is as the 
 fpace dire5lly, and time reciprocally \ or as the fpace 
 divided by the time. And if the time be the fame, 
 the velocity is as the fpace defcribed. And if the 
 fpace be the fame, the velocity is reciprocally as tht 
 time of defcription. 
 
 PROP.
 
 Sea. I. OFMOTION. f 
 
 PROP. IV. 
 
 The motion generated by any momentary force, or 
 hy a Jingle impilfe, is as the force that generates it. 
 
 For if any force generates any quantity of mo- 
 tion; double the force will produce double the 
 motion ; and treble the force, treble the motion, 
 and fo on. If a body ftriking another, gives it any 
 motion, twice that body flriking the fame, with the 
 fame velocity, will give it twice the motion, and i<i 
 the motion generated in the other will be as the 
 force of percuflion. 
 
 Cor. I . Hence the forces are in the compound ratio 
 of the 'velocities and quantities of matter. 
 
 For (Prop. II.) the motions are as the quanti- 
 ties of matter multiplied by the velocities. 
 
 Cor. 2. The 'velocity generated, is as the force di~ 
 really, and quantity of matter reciprocally There- 
 fore if the bodies are equal, the velocities are as the 
 forces. And if the forces are equal, the 'velocities are 
 reciprocally as the bodies. 
 
 Cor. 3. The quantity of matter, is as the force di- 
 rectly, and velocity recip ocally. And therefore if the 
 velocities be equal, the bodies are as the forces. And 
 if the forces be equal, the bodies are reciprocally as 
 the velocities. 
 
 PROP. V. 
 
 The quantity of motion generated, by a conftant and 
 uniform force, is in the compound ratio of the force 
 and time of aSitng. 
 
 For the motion generated in any given time will 
 
 be proportional to the force that generates it % 
 
 B 4 and
 
 fi GENERAL LAWS 
 
 and in twice that time, the motion will be double 
 by the fame force -, and in thrice that time it will 
 be treble ; and fo every part of time adds a new 
 quantity of motion equal to the firft -, and there- 
 fore the wliole motion, will be as the force, and 
 the whole time of afting. 
 
 Cor. I . The motion lofi in any time^ is in the corn- 
 found ratio of the force and time. 
 
 Cor. 2. The velocity generated (or dejlroyed) in 
 any titne^ is as the force and time dire£ily, and quan- 
 tity of matter reciprocally. The fame is true of the 
 incrcafe or decreafe of 'velocity. 
 
 For the motion, that is (Prop. II.) the body mul- 
 tiplied by the velocity, is as the force and time. 
 And therefore the velocity is as the force and time 
 djredly, and the body reciprocally. 
 
 Cor. 3. Hence if the force be as the quantity of 
 matter., the velocity is as the time. Or if the force 
 and quantity of matter be given., the velocity is as the 
 time. 
 
 And if the time and quantity of matter be given, 
 the velocity is as the force. 
 
 And if the force and time be given., the velocity is 
 reciprocally as the jnatter. 
 
 Cor. 4. The time is as the quantity of matter and 
 velocity dirc5ily, and the force reciprocally. Therefore, 
 
 If the force and velocity be given, the time is as 
 the quantity of matter. 
 
 If the quantity of matter and velocity be given, 
 the time is reciprocally as the force. 
 
 Cor. 5. The force is as the quantity of matter and 
 •velocity diretlly, and the time reciprocally. Whence, 
 
 If the velocity is at the time, or if the velocity and 
 time be given, then the force is as the quantity of matter. 
 
 And
 
 Sea. I. OFMOTION. 9 
 
 And if the velocity and quantity of matter be given^ 
 the force is reciprocally as the time. 
 
 Cor. 6. ^he quantity of matter is as the force and 
 time dire^ly, and the velocity reciprocally. There- 
 fore^ if the force and time be given, the quantity of mat- 
 ter is reciprocally as the velocity. 
 
 Cor. 7. Hence alfo if the body be given, the velo- 
 city is in the compound ratio of the force and time. 
 
 And if the force be given, the time is in the com- 
 pound ratio of the matter and velocity, or as the 
 quantity of motion. 
 
 P R O P. VI. 
 
 If a given body is urged by a conflant and uniform 
 force \ the fpace defcribed by the body from the be- 
 ginning of the ^notion, vjill be as the force and fquare 
 of the time. 
 
 Suppofe the time divided into an infinite num- 
 ber of equal parts or moments. Then in each of 
 thefe moments of time, the fpace defcribed (Prop. 
 in. Cor. 2.) will be as the velocity gained; that is, 
 (by Cor. 7. Prop. V.) as the force and time from 
 the beginning. And the fum of all the fpaces, 
 or the whole fpace defcribed, will be as the force 
 and the fum of all the moments of time from the be- 
 ginning. Therefore put / ~ the whole time, and 
 the whole fpace defcribed will be as the force and 
 fum of the times i, 2, 3, 4, &;c. to /. But the 
 fum of the arithmetic progrefPion i -1-2 4-3+4 
 
 ^4-1 
 . . . to / zz X t zz ^ tt, becaufe t is infinite 
 
 2 
 
 or confifls of an infinite number of moments. There- 
 fore the whole fpace defcribed will be as the force 
 and 4- tt\ that is, (becaufe 4- is a given quantity), 
 as the force and the fquare of the time of defcription. 
 
 Con
 
 lo GENERAL LAWS 
 
 Cor. I. If a body is impelled by a confiant and 
 
 ttniform force •, the /pace defcribed from the btginning 
 
 of the motion^ is as the velocity gained^ and the time 
 
 of moling. 
 
 For the fpace is as the force and fqiiare of the 
 
 time, or as the force X time X time. But (Cor. 7. 
 
 Prop. V.) the force X time is as the velocity; 
 
 therefore the fpace which is as the force X time X 
 
 time, is as the velocity X time. 
 
 Cor. 2' If a body urged by any conjiant and uni- 
 form force^ defcribes any fpace j // will defer ibe twice 
 that fpace in the fame time, by the velocity acquired. 
 
 For the fum of all the fpaces defcribed by that 
 force, 1+2+3 ^^* ^° ^> ^^^ fhewn to be ^ tt. 
 But the fum of ail the fpaces defcribed by the laft 
 velocity, will be / + / + ^ &c. to / terms, whofe 
 fum is tt. But tt is double to 4 tt ; that is, the 
 fpace defcribed by the laft velocity, is double the 
 fpace defcribed by the acceleratmg force. 
 
 Cor. 3. Univerfally in all bodies urged by any con- 
 ft ant and uniform forces -, the fpace defc^ ibed is as the 
 force and fquare of the time dire£lly, and the quanti- 
 ty of matter reciprocally 
 
 For (Cor i.) the fpace is as the time and ve- 
 lo ity. But (Prop V. Cor 2.) the velocity is uni- 
 verfally as the force and time dire<5tly, and quan- 
 tity of matter reciprocally. Therefore the fpace is 
 as the fquare of the time and the force diredly, and 
 matter reciprocally ; whence, 
 
 Cor. 4. The froduSl of the force ^ and '-lare of 
 the time ^ is as the produol of the body and fpace de' 
 fcribed. 
 
 Cor. 5. The product of the force and time, is dS" 
 the produfl of the quantity of matter and velocity. 
 For (Prop. V.) the produdt of the force and 
 
 time.
 
 Seft. I. O F M O T I O N. n 
 
 time, is as the motion ; that is, as the body and Fig. 
 velocity. 
 
 Cor. 6. 7he produ5f of the body, and fquare of 
 the velocity, is as the product of the force and the 
 fpnce defcribed. 
 
 For (Cor. 5.) the produd of the body and ve- 
 locity, is as the force and time. Therefore, the 
 body X velocity fquare, is as the force X time X 
 velocity ; but time X velocity is as the fpace (by 
 Prop. III.) i therefore body X velocity fquare is as 
 the force X fpace. 
 
 Scholium. 
 
 If any quantity or quantities are given, they muft 
 be left out. And fuch quantities as are propor- 
 tional to each other muft be left out. For exam- 
 ple, if the quantity of matter be always the fame ; 
 then (Cor 3.) the fpace defcribed is as the force 
 and fquare of the time. And if the matter be 
 proportional to the force, as all bodies are in ref- 
 ped to their gravity •, then (Cor. 6.) the fpace de- 
 scribed is as the fquare of the velocity. Or if the 
 fpace defcribed be always proportional to the bo- 
 dy ; then (Cor. 6.) the force is as the fquare of the 
 velocity. Again, if the body be given, then (Cor. 
 4.) the fpace is as the force and fquare of the time. 
 And if both the quantity of matter and the force 
 be given j the fpace defcribed is as the fquare of 
 the time. And fo of others. 
 
 PROP. VIT. 
 If ABCD he a -parallelogram \ and if a body at j; 
 A, he a£led upon feparately by two forces, in the di' 
 regions AB and AC, ivhich would caufe the body to 
 he carried thro* the fpaces AB, AC in the fame time. 
 Then both forces aSiing at once, will caufe the body ta 
 be carried thro' the diagonal AD of the parallelogram. 
 
 Let the line AC be fuppofed to move parallel toi 
 
 itfelf;
 
 12 GENERAL LAWS 
 
 Fig. itfelf ; whilft the body at the fame time moves 
 I. from A, along the line AC or bg^ and comes to d 
 at the fame inftant, that AC comes to bg. Then 
 fince the lines AB, AC, are defcribed in the fame 
 time -, and A^, Ai, are alfo defcribed in the fame 
 time. Therefore, as the motions are uniform, it 
 will be, A^ : ^^ : : AB : BD i and therefore AJD 
 is a flreight line, coinciding with the diagonal of 
 the parallelogram. 
 
 Cor. I. The three forces in the direEiions h^^ AC, 
 AD, are refpe5Iively as the Imes AB, AC, AD. 
 
 Cor. 2. Jny Jingle force AD denoted by the diago- 
 nal of a parallelogram^ is equivalent to two for ces de- 
 noted by the fides AB, AC. 
 
 Cor. 3. And therefore any fngle force AD may he 
 refohed into two forces^ an infinite Jiumber of waySy 
 by drawing any two lines AB, BD, Jor their quanti- 
 ties and dire^iions. 
 
 Scholium. 
 This practice of finding two forces equivalent to 
 one, or dividing one force into two ; is called the 
 compoCttion and refoludon of forces. 
 
 PROP. VIII. 
 
 .'■ / 
 .2. If three forces A, B, C, \cep one another in equi- -^ 
 librio ', they will be proportional to three fides of a 
 iriancle^ d' awn parallel to their fever al directions ^ 
 DI, CI, CU. 
 
 Produce AD to I, and BD to H, and compleat 
 the parallelogram DICH •, then (Prop. VII.) the 
 force in diredlion DC, is equal to the forces DH, 
 DI, in the diredions DH, DI. Take away the 
 force DC and putting the forces DH, DI equal 
 
 thereto ^
 
 H 
 
 Sea. I. OF MOTION, 
 
 thereto-, and the equilibrium will ftill remain Fig. 
 Therefore (Ax. lo.) DI is equal to the force A op- 2. 
 pofite to it •, and DH or CI equal to its oppofite 
 force B. And as CD reprcfents the force C, the 
 three forces A, B, and C will be to one another as 
 DI, CI, and CD. 
 
 Cor. I. Hence if three forces ailing againfi one 
 another., keep each other in equilibrio ; thefe forces 
 will he refpe^ively as the three ftdes of a triangle 
 drawn perpendicular to their lines of dire^ion ; or 
 making any given angle with them., on the fame fide. 
 
 For this triangle will be fimilar to a triangle 
 whofe fides are parallel to the lines of direftion. 
 
 Cor. 2. If three aolive forces A, B, C keep one 
 another in equilihio ; they will be refpe5iively as the 
 fines of the angles^ which their lines of direHion pafs 
 through. 
 
 For A, B, C are as DI, CI, CD; that is, as 
 S.DCI, S.CDI, and S.DIC. But S.DCI = S.CDH 
 = S.CDB. And S.CDI zi S.CDA. Aifo S.DIC 
 =: S.BDI - S.BDA. 
 
 Cor. 3. If ever fo many forces a^ing againfi one 
 another, are kept in eiuilihrio, by thefe actions •, they 
 may he all reduced to two equal and oppofite ones. 
 
 For any two forces may, by compofition, bereduced 
 to one force a6ting in the Ume plane. A.nd this 
 laft force, and any other, may likewife be reduced 
 to one force afting in the plane of thefe ; and fo 
 on, till they all be reduced at laft to the adion of 
 two equal and oppofite ones. 
 
 PROP. IX. 
 
 If a body impinges or a^s againjl any plain furface ; 3. 
 it exerts its force in a line perpendicular to that furface. 
 
 Let the body A moving in direfticn AB, with a 
 given velocity, impinge on the imooth plain FG at 
 
 the
 
 14 GENERAL LAWS 
 
 fig. the point B. Draw AC parallel, and BC perpen- 
 3. dicular to FG •, and let AB reprefentthe force of 
 the moving body. The force AB is, by the refo- 
 lution of forces, equivalent to AC and CB. The 
 force AC is parallel to the plain, and therefore has 
 no effect upon it; and therefore the furface FG 
 is only afted upon by the force CB, in a diredtion 
 perpendicular to the furface FG. 
 
 Cor. I . If a body impinges upon another body with 
 a given velocity, the qiiantity of the firoke is as the 
 fine of the angle of incidence. 
 
 For the abfolute force is AB, and the force 
 . aaing on the furface FG is CB. But AB : CB : : 
 rad : S.CAB or ABF. 
 
 Cor. 1. If an elajlic body A impinges upon a hard 
 $r elajlic plane FG \ the angle of reflexion will be equal 
 to the angle of incidence. 
 
 For if AD be parallel to FG -, the motion of A 
 in direftion AD parallel to the plane, is not at all 
 changed by the llroke. And by the elaflicity of 
 one or both bodies, the body A is refledled back to 
 AD in the fame time it moved from A to B ; let 
 it pafs to D ; then will AC zz CD, being defcribed 
 in equal times -, confequently the angle ABC — 
 angle CBD ; and therefore the angle DBG — an- 
 gle ABF. 
 
 Cor. Q,. If a non-elajlic body Jlrikes another non- 
 elajlic body ; it lofes but half the motion^ that it would 
 lofe^ if the bodies were elajlic. 
 
 For non-elaftic bodies only Hop, without recede- 
 ing from one another j but elailic bodies recede 
 with tlie fame velocity. 
 
 PROP,
 
 
 Sea. L O F M O T I O N. 15 
 
 PROP. X. 
 
 ^be fum of the motions of two or more bodies^ 
 in an'' dire5lion towards the fame part^ cannot be 
 ibanged by any aSion of the bodies upon each other. 
 
 Here I reckon progrefllve motions afErmative; 
 and regrefllve ones negative, and to be deduced 
 out of the reft to get the fum. 
 
 1 . If two bodies move the fame way -, fmce ac- 
 tion and re-a<5tion are equal and contrary, what one 
 body gains the other lofes •, and the fum remains 
 the fame as before. And the cafe is the fame, if 
 there were more bodies. 
 
 2. If bodies ftrike one another obliquely ; they 
 will a6l on one another in a line perpendicular to 
 the furface afted on. And therefore by the law of 
 adion and re-a6lion there is no change made in that 
 diredion. 
 
 3. And in a diredion parallel to the ftriking fur- 
 face, there is no aftion of the bodies, therefore the 
 motion remains the fame in that direction, "Whence 
 the motions will remain the fame in any one line of 
 diredlion. 
 
 Cor. I. Motion can neither be increafed nor de- 
 creafed^ confid?red in any one direBion -, but muft re- 
 main invariably the fame for ever. 
 
 This follows plainly from this Prop, for what 
 motion is gained to one (by addition), is loft to 
 another body (by lubtradion) •, and fo the total fum 
 remains the fame as before. 
 
 Scholium. 
 
 This Prop, does not include or meddle with fuch 
 motions as are eftimated in all direftions. For 
 upon this fuppofition, motion may be increafed or 
 
 decreafed
 
 |6 GENERAL LAWS. 
 
 Fig. decreafed an infinite number of ways. For example, 
 if two equal and non-elaftic bodies, with equal 
 velocities, meet one another ; both their motions 
 are deftroyed by the ftroke. Here at the beginning 
 of the motion, they had both of them a certain 
 quantity of motion, but to be taken in contrary 
 diredtions ; but after the ftroke they had none. 
 
 PROP. XI. 
 
 ^he motion of bodies included in a given fpace^ is 
 the fame, whether that fpace Jiands fill •, or moves 
 uniformly in a right line. 
 
 For if a body be moving in a right line, and any 
 force be equally imprefled both upon the body and 
 the line it moves in ; this will make no alteration 
 in the motion of the body along the right line. 
 And for the fame reafon, the motions of any num- 
 ber of bodies in their feveral diredions will ftill re- 
 main the fame •, and their motions among themfelves 
 will continue the fame, whether theincluding fpace is 
 at reft, or moves uniformly forward. And fince the 
 motions of the bodies among themfelves \ that is, 
 their relative motions remain the fame, whether 
 the fpace including them be at reft, or has any uni- 
 form motion. Therefore their mutual a6^ions upon 
 one another, muft alfo remain the fame in both 
 cafes. 
 
 SECT,
 
 [^7 J 
 
 S E C T. 11. ^'^' 
 
 The fierpendictdur defcent of heavy id- 
 dies, their defcent upo?i i?2C lined Pla?2es^ 
 and i?t Curve Surfaces, The Motion 
 of Pendulums^ 
 
 PROP. xir. 
 
 tr'H E velocities of bodies^ falling freely by their own 
 weighty are as the times of their falling from refi, 
 
 For fince the force of gravity is the fame in all 
 places near the earth's furface (by Poft. 3.), and 
 this is the force by which bodies defcend. There- 
 fore the falling body is urged by a force which ads 
 conftantly and equally ; and therefore (by Cor. 3. 
 Prop. V.) the velocity generated in the falling bo- 
 dy in any time, is as the time of falling. 
 
 Cor. I . If a body be thrown direSlly upwards with 
 the fame velocity it falls with \ it will lofe all its mo- 
 tion in the fame time. 
 
 For the tame adlive force will deftroy as much 
 motion in any time, as it can generate in the fame 
 time. 
 
 Cor. 2. Bodies defending or afcendinf gain or lofe 
 eqiial velocities in eqtial times. 
 
 PROP. XIH. 
 
 In bodied falling freely by their own weight \ the 
 fpaces defcrihed in falling front fejl^ are as the fquares 
 of the times of falling. 
 
 For fmce gravity is fuppofed to be tfie farne \n 
 
 all places near the earth. Therefore the falling 
 
 C body
 
 iS FALLING BODIES. 
 
 Fig. body will be acted on by a force which is con{lant> 
 ly the famej and therefore (by Prop. VI.) the 
 fpaces defcribcd, from the beginning of the mo- 
 tion, or fince their falling from reft, will be as the 
 fquares of the times of falling. 
 
 Cor. I. 'The fpaces defcribed by falling bodies are 
 as the fquares cf the velocities. 
 
 For (Prop. XII.) the velocities are as the times 
 of falling;. 
 
 Cor. 2. The fpaces defcribed by falling bodies^ are 
 in the compound ratio of the times, and the velocities 
 acquired by falling. 
 
 Cor. 2- If a body falls through any fpace, and 
 7nove aftervsards 'with the velocity gained in falling ; 
 it will defcribe twice that f pace in the time of its falling. 
 
 Cor. 4. A body projeBed upward with the velocity 
 it gained in falling, will afcend to the fame height ii 
 fell from. 
 
 Scholium. 
 
 All thefe things would be true if it was not for 
 the refiitance of the air, which will retard their 
 motion a little. In very fwift motions, the refif- 
 tance of the air has a very great effed in deltroy- 
 ing the motions of bodies. 
 
 PROP. XIV. 
 
 4. If a heavy body W be fiiftained upon an inclined 
 plane AC, Z'V a power F atiing in direction WF pa- 
 rallel to the plane \ and // A B be -parallel to the hori- 
 zon and EC perp. to it. Then if the length AC de- 
 notes the weight of the body, the hight CB will denote 
 the power at F which fu ft ains itt and the bafe AB th^ 
 preffure againfi the plane. 
 
 Draw BD perpendicular to AC, then CB v/ill be 
 the diredion of gravity, DC parallel to WF will 
 
 be
 
 Sea. II. INCLINED PLANES. 19 
 
 be the direction of the force at F, and DB the di- Fig, 
 region of the prefTure (by Prop. IX ). Therefore 4. 
 (by Prop. VIlI.) the weight of the body, the 
 power at F, and the preliV.re •, will be refpeclively 
 as BC, CD and DQ. But the triangles ABC, BDC 
 are fimilar, and therefore BC, CD and DB are rcf- 
 pedively as AC, CB and BA. Therefore the 
 weight, power and preilure, are as the lines A.Cj 
 CB and AB. 
 
 Cor. I . The iveight of the hod)\ the parer F that 
 fujiains it on the flane^ and the ■preifitre againj} the 
 plane \ are refpeoJively as radius, the fine and cqfne 
 of the planes elevation above the horizon. 
 
 For AC, CB and AB are to one another, as 
 radiusj fine of CAB, and fme of ACB. 
 
 Cor. 2. The pczver that urges a body W dotvn the 
 
 CB 
 
 inclined plane ts r= — r-r x "wetght of W. Hence, 
 
 Cor. 3. If a pripnatic body whofe length is AC 
 lie upon the inclined plane AC •, // is urged do'j^n the . 
 plane ivilh a force equal to the weight of the prifni 
 of the length CB. 
 
 Cor. 4. If there he two planes of the fame bight ^ 
 and two bodies be Iriid en them proportional to the 
 lengths of thefe planes ; the tendency down the planes 
 will be equal in both todies. 
 
 PROP. XV. 
 
 If AC be an inclined plane, AB the horizon^ BC 5, 
 perp. to AB. And if W he a heavy body up n the 
 plane, which is kepc there by the power P aclin^ in 
 direfiion VvP. Draw BDE perp. to \Vp. fhe^i 
 the weight \V, the power P, the prefure againft the 
 plane \ will be refpei^tively as xAB, DB and AD. 
 
 For AB being a horizontal plane is perpendicu- 
 lar to the action of gravity ; and Y^D is perp. to 
 C I the
 
 20 INCLINED PLANES. 
 
 Fig. the diredlion of the power P; and AD is the 
 5. plane, which is perp. to the direction of the pref- 
 fure againft that plane. Therefore (Cor. i . Prop. 
 Vlll.) the weight of the body, the power P, and 
 the prefTure ; are as AB, BD and AD. And if 
 the direftion of the power WP be under the plane, 
 the proportion will be the fame, as long as BD is 
 perpendicular to WP. 
 
 Cor. I. 'The weight of the body \V, is to the^ower 
 P that fujiains it : : as cofine of the angle of traftio?i 
 CWP : to the fine of the plane's elevation CAB. 
 
 For the weight : power : : AB : BD : : S.ADB 
 or WDE : S.BAD : : cof DWE or CWP : S.BAC, 
 where the angle CWP made by the plane and di- 
 redtion of the power is called the angle of Tradion. 
 
 Cor. 2. Hence it is the fame thing as to the power 
 
 and weighty whether the line of direction is above or 
 
 below the plane^ provided the angle of traction be the 
 
 Jame, For an e^ital power will fuflain the weight in 
 
 both cafes. 
 
 Cor. 3. The weight of the body : is to the preffure 
 dgainft the plane : : as the cofine of the angle of trac- 
 tion CWP : to the cofine of BNP, the dire^ion of 
 the power above the horizon. 
 
 For the weight W : prefTure : : AB : AD : : 
 S.ADB : S ABD : : S.ADE : : S.NBE : : cof. 
 EWD or PWC : cof BNP. 
 
 Cor. 4. Hence the -preffure againfi the plane is 
 greater when the direBion of the power is below the 
 plane, the weight remaining the fame. 
 
 Scholium. 
 
 Altho' the power has the fame proportion to the 
 weight, when the angle of traftion is the fame ; 
 whether the direction of the power be above or 
 
 below
 
 Sea. II. INCLINED PLANES. 21 
 
 below the plane. Yet, fince the preflure upon the Fig. 
 plane is greater, when the line of dired:ion is be- 5. 
 low the plane. Therefore in pra6tice, when a 
 weight is to be drawn up hill, if it is to be 
 done by a power whofe direction is below the plane, 
 the greater prefllire in this cafe will make the car" 
 riage fink deeper into the earth, &c. and for that 
 reafon will require a greater power to draw it up, 
 than when the line of diredion is above the plane. 
 
 PROP. XVI. 
 
 If a weight W titon an inclined plane AC, be in 6. 
 equilibria with another weight P hanging freely ; then 
 if they be fet a moving^ their perpendicular velocities 
 in that place., will be reciprocally as their quantities of 
 matter. 
 
 Take WA a very fmall line upon the plane AC; 
 draw AE parallel to the horizon, and BC perp. to 
 it. Draw AF, and WR, BE perpendicular to it ; 
 and WT, DV perp. to AB. Let W defcend thro* 
 the fmall line WA upon the plane, then P will af- 
 cend a hight equal to AR perpendicularly ; and 
 \VT will be the perpendicular defcent of W. The 
 triangles AWR and ADE are fimilar ; and like- 
 wife the triangles AWT and ADV. Therefore 
 WT : DV : : AW : : WD : : WR : DE. And 
 alternately, WT : WR : : DV : DE ; and WR : 
 AR : : DE : AE ; therefore WT : AR : : DV : 
 AE : : (by the fimilar triangles DBV and AEB) 
 D^ : AB : : (Prop. XV.) power P : weight W. 
 
 Cor. I . If any two bodies be in equilihrio upon two 
 inclined planes ; their perpendicular velocities will be 
 reciprocally as the bodies. 
 
 Cor. 2. If two todies fuftain each other in equili- 
 hrio^ on any planes •, the prodti^ of one body X by its 
 
 C 3 ' perp.
 
 28 INCLINED PLANES. 
 
 Fig. perp. velocity^ is equal to the p'sduH of the other ho' 
 6. dy X by Its perp. vtlcdty. 
 
 PROP. XVII. 
 
 7* If a heaiy hody runs down an inclined plane CA -, 
 the velocity ;/ acquires in any time, moving from reft •, 
 is to the velocity acquired by a body falling perpendi- 
 cularly in the fame time i c.s the hight of the plane CB, 
 to its k'ngth CA. 
 
 The force by which a body endeavours to defcend 
 on an inclined plane, is to its weight or the force 
 of gravity -, as CB to CA (by Prop. XIV.). And 
 as thefe forces always remain tlie fame, therefore 
 (Cor. 2. Prop. V.) the velocities generated will be 
 as thefe forces, and the times of aCling, directly ; 
 and the bodies reciprocally. And fmce the times 
 of ading, and the bodies are the fame in both 
 cafes, the velocities generated will be as thefe 
 forces ; that is, as the hight of the plane CB. to 
 its length CA. 
 
 Cor. I. The velocity acquired by a body running 
 dcivn an inclined plane, is as the time of its moving 
 from reft.. 
 
 Cor. 1, Jf a hoJy is ihrcivn up an inclined plane, 
 ivith the velocity it acquired in defending ; it will lofe 
 cll its motion in the fame time. 
 
 PROP. XVIII. 
 
 •m^ Jf a heavy body defends down an inclined plane 
 CA \ the fpace it defcribes frc?n the beginning of the 
 motion, is to the fpace defctibed by a body falling per- 
 peyidiculcrly in the fame time ; as the hight of the 
 -plane CB, to its length CA. 
 
 For the force urging the body down the plane is 
 to the force of its gravity, as CB to CA (by Prop. 
 
 ^ XIV.),
 
 Sea. IL INCLINED PLANES. 23 
 
 XIV.), which forces remain conftandy the fame. Fig. 
 And lince (Prop. XVII.) the velocities generated 7, 
 in equal times on the plane, and in the perpendi- 
 cular, are conilantly as CB to CA ; the fmall par- 
 ticles of fpace delcribed with thefe velocities, in 
 all the infinitely fmall portions of time, will flill 
 be in the fame ratio; and therefore the fums of 
 all thefe fmall fpaces, or the whole fpaces defcribed 
 from the beginning, vvrill be in the fame conftant 
 ratio of C 3 to CA. 
 
 Cor. I . The fpace defcribed by a body falling do-wn 
 an inclined plane ^ in a given iime, is as the fine of 
 the planers elevation. 
 
 For if CB be given, and aifo the perp. defcent ; 
 that fpace will be reciprocally as CA, or diredly 
 as S.CAB. 
 
 Cor. 2 . 'The fpaces defcribed by a body defending 
 from reft., dozvn an inclined plane, are as the fquares 
 of the times. 
 
 Cor. 3. Jf BD be drawn perp. to the plane C A *, 
 then in the time a body falls perpendicularly thro' the 
 bight CB, another body will defcend thro* the fpace 
 CD upon the plane. 
 
 For by fimilar triangles CA : CB : : CB : CD. 
 
 PROP. XIX. 
 
 If AC is an inclined plane ^ the time of a body^s 7. 
 defending thro* the plane CA, is to the time of fall- 
 ing perpendicularly thro* the hight of the plane CB, 
 AS the length of the plane CA to the. hight CS. 
 
 For if BD is perp. to CA, then (Cor. 2. Prop. 
 XVIII.) fpace CD : fpace CA : : fquare of the 
 time in CD : fquare of the time in CA : : that is, 
 as the fquare of the time of defcending perpendi- 
 
 C 4 culariy
 
 24 INCLINED PLANES. 
 
 ¥\cr. cukrlv in CB (Cor. 3. laft) : fquarc of the titiK- in 
 
 7° C^. But CD : CA : : CB^ : CA\ Therefore 
 
 CB' : CA' : fquare of the time in CB : fquare of 
 
 the time in CA. And CB : CA : : time in CB : 
 
 time in CA. 
 
 Cor. I. If a l^ody he throruDn upwards en the plane 
 with the velocity acquired ir. defcending\ it ivill in 
 an epial time afcend to the fame hight. 
 
 Cor. 2 The times ivherein different planes^ of the 
 fame hight, are paffed over ; are as the lengths of the 
 planes. 
 
 Let the planes be CA, CF. Then time in AC : 
 time in CB : : CA : CB. And the tim,e in CB : 
 time in CF : : CB : CF. Therefore ex equo, time in 
 AC : time in CF : : CA : CF. 
 
 PROP. XX, 
 
 o Jf a body falls dcivn an inclined plane, it acquires 
 
 the fame velocity as a body falling pei-pendicularly 
 thro' the hight of the plane. 
 
 Let the body run down the plane CA whofe 
 hight is CB. Draw PF parallel to AB, and infi- 
 nitely near it. Then the velocities in DA and FB, 
 may be looked upon as uniform. Now fProp. 
 XIX.) the times of defcribing CA and CB, will be 
 as CA and CB. l^ikewifc the times of dcferibing 
 CD and CF, will be as CD and CF •, that is, as 
 CA and CB. And by divifion, the difference of 
 the times, or the times of defcribing DA and FB, 
 will alto be as CA and CB •, that is, as DA and 
 FB. Rut (from Prop. III.) the velocities are equal 
 when the fpaces are as the times of defcription. 
 Therefore velocity at A is equal to the velocity 
 at B. 
 
 Cor.
 
 Sea. II. INCLINED PLANES. 25 
 
 Cor. I . The velocities acquired^ by bodies defcend- Fig. 
 ing on any planes^ from the fame bight to the fame ho- 8. 
 rizontal line, are equal. 
 
 Cor. 2. If the velocities be equal at any two equal 
 altitudes D, F ; they ivilt be equal at all other equal 
 altitudes A, B. 
 
 Cor. 3. Hence alfo^ if fever al bodies be jnoving in 
 different dire^ions^ thro^ any fpace contained between 
 two parallel planes -, and be a5led on by any force., 
 which is €qual at equal dijlances from either pla7te. 
 Then if their velocities be equal at entering that 
 fpace ; they will alfo be equal at emerging out of it. 
 
 For dividing that fpace into infinitely fmall 
 parts by parallel planes. Then the force between 
 any two planes may be fuppofed uniform -, and fup- 
 pofing DP\ AB to reprefent two of thefe planes, 
 then (by Cor. 2.) the velocities at D and F' being 
 equal-, the velocities at A and B will be equal; 
 that is, the velocities at entering the firft part of 
 Ipace being equal, the velocities at emerging out 
 ot it, or at entering the fecond fpace will be equal. 
 And for the fame reafon the velocities at entering 
 the fecond fpace being equal, thofe at emerging 
 out ot it into the third, will be equal. And con-^ 
 fequently the velocities at entering into, and emerg- 
 ing out of the third, fourth, fifth, &c. to the laft 
 \vill be equal refpedively. 
 
 PROP. XXL 
 
 If a body falls from the fame hight., thro' am 9. 
 number of contiguous planes AB, BC, CD ; it 
 will at lajl gain the fame velocity as a body falling 
 perpendicular^ from the fame hight. 
 
 Let FH be a ho'ir.ontal line, FD perp. to it. 
 Produce the planes BC, DC to G and H. Then 
 
 (Cor.
 
 26 INCLINED PLANES. 
 
 Fig. (Cor. 1. Prop. XX.) the velocity at B is the lame 
 c^T whether the body defcend thro' AB or GB. And 
 therefore the velocity at C v/ill be the fame, whe- 
 ther the body defcends thro' ABC or thro' GC, 
 and this is the fame as if it had defcended thro* 
 HC. And confequently it will have the fame ve- 
 locity at D, in delcending thro' the planes AECD, 
 as in defcending thro' the plane HD ; that is, 
 (Prop. XX.) as it has in defcending thro' the per- 
 pendicular FD. 
 
 Cor. I. Hence a body defcending along cny curve 
 furface, ilUI acquire the fame velocity^ as if it fell 
 ferpendiculcn ly thro* the jame hight. 
 
 For let the number of planes be increafed, and 
 their length diminifhed ad infinitum, and then 
 ABCD will become a curve. And the velocity 
 acquired by defcending thro' thefe infinite planes j 
 that is, thro' the curve ABCD, will be the fame 
 as in falling perpendicularly thro' FD. 
 
 Cor. 7.. If a hcdy defcends in a curie ^ and ano- 
 ther defcends -perpendicularly from the fame hight. 
 S^beir velocities "will be equal at all equal altitudes. 
 
 Cor. ^. If a bcdy^ after its defcent in a curve y 
 Jhould be dire^ed upwards with the 'velocity it had 
 gained ; it will afcend to the fame hight frcm which 
 it fell. 
 
 For fince gravity acts with the fame force whe- 
 ther the body afcends or defcends, it will dellroy 
 the velocity in the afcent, in the fame time it did 
 generate it in the dclcent. 
 
 Cor. 4. The %'elocity of a bcdy defcending in any 
 furve, if as the fquare root of the hight fallen from. 
 
 For it is the fame as in talhng perpendicularly ; 
 and in falling perpendicularly, it is as the fquare 
 root of the hight., 
 
 Cor.
 
 Sea. 11. INCLINED PLANES. 27 
 
 Cor. ^. If a hody^ in moving thro" afiy /pace ED, Fig, 
 he a£led en uniformly hy any force -, its velocity at 9. 
 emerging out of it at D, will be equal to the fquare 
 root of the fum of the fquares of the vtloctty at E 
 in entering of ity and of the velocity acquired in fall- 
 ing from rejl thro"" that fpace ED. And this holds 
 whether the body moves perpendicularly or obliquely. 
 
 For let the body enter the fpace ED at E, with 
 the velocity acquired in falling thro' FE. Then 
 (Prop. XIU. Cor. i.) the fquare of the velocity at 
 E will be as FE-, and the fquare of the velocity at 
 D, as FD-, and the fquare of the velocity at I) 
 falling from E, will be as ED. But FD = FE 
 + ED ; therefore the fquare of the velocity at 
 D (falling thro' FD) zz fquare of the velocity at 
 E + fquare of the velocity at D (falling thro' ED). 
 And (Cor. i. of this Prop.) the velocity will be 
 the fame whether the body defcends perpendicu- 
 larly or obliquely. 
 
 PROP. XXII. 
 
 ^he times of bodies defcending thro* two fimilar 'O. 
 parts of Jimilar curves^ placed alike^ are as the 
 'fquare roots of their lejigths. 
 
 Let A BCD and abed be two fimilar curves, and 
 fuppofe BC and be to be inhnitely imall, and fimi- 
 lar to the whole ; that is, fo that BC : ^f ; : AB : 
 ab. Draw FA parallel to the horizon, and HB, 
 hb perp. to it. Then if two bodies defcend from 
 A and a (Cor. 4. Prop. XXI.) the velocities at B 
 and b will be as v/HB and y/hb •, that is, as v/AB 
 and \/aby becaufe AH, ab are fimilar parts. 
 Therefore (Prop. III. Cor. i.) the times of de- 
 
 BC ^^ , . 
 
 fcribins; BC and be. are as > — : and , , •, that is, 
 ^ VAB \/<^^ 
 
 ' as
 
 25 INCLINED PLANF^S. 
 
 I'ig. AB J ^^ — 
 
 JO. as "TTg a"a ~7ab ^^ ^^ \/-A^ and s/ah; that is> 
 
 as v/AD and ^ad^ becaufe the curves are fimi- 
 larly divided in B and h. After the fame manner 
 the times of defcribing any other two fimilar parts 
 as BC, bc^ will be as v/AD and '^ad. Therefo:c 
 by compofition the times of defcribing all the BC's, 
 and all the bc'^ will be as ^/AD and *yad. That 
 is, the time of delcribing the curve AD to the" 
 time of defcribing the curve ad^ is as y/AD to 
 ^ad. 
 
 Cor. If t'uco bodies defcend thro* two fimilar curies 
 ABD, and abd \ the axes of the curves FD, F^ art 
 as the fqiiares of the times of their defcending. 
 
 For v/FD : Wd : : n/aD : a/^ : : time of 
 defcending thro' ABD : time of defcending thro' abd. 
 And FD, Fd, are as the fquares of the times. 
 
 PROP. XXIII. 
 
 I y ; ^ body IV ill defcend thro* any chord of a circle^ in 
 the fame time that another defcends J)erfendicularl^ 
 thro* the diameter. 
 
 Draw the diameter AB perpendicular to thehori^ 
 zon, and the cords CA, CB. Then fmce BC is 
 perpendicular to AC, therefore (Prop. XVIIT. 
 Cor. 3.) the time of defcending thro' the cord AC 
 is equal to the time of falling thro' AB. 
 
 Draw CD parallel to AH, and DB parallel to 
 CA, then is CI) equal to AB. And by reafon of 
 the parallels, the angle DBC zz angle EC A zz 4 
 right angle. Tlien fince DB is perp. to CB, there- 
 fore (Cor. 3. Prop. XVIII.) a body will defcend 
 thro' the inclined plane CB, in the fame time that 
 another falls thro' CD, or which is the fam.e thing, 
 thro' its equal AB. 
 
 Cor.
 
 Sea.II. INCLINED PLANES. 29 
 
 Cor. I. Hence the times of defcending thro* alt the Fig. 
 cords of a circle drawn from A or B, are equal 1 1, 
 among themfehes. 
 
 Cor. 2. 'I'he velocity gained hy falling thro^ the 
 cord CB, is as its length CB. 
 
 For the velocity gained in falling thro' CB is 
 the fame as is gained by falling thro' EB -, and 
 that velocity is to the velocity gained by fallino- 
 thro' AB, asv/BE to v/AB (by Cor. i. Prop! 
 XIII.) i that is, as BC to BA. Therefore if the 
 given velocity in falling down AB be reprefented 
 by AB. The velocity gained in falling down CB 
 will be reprefented by CB ; and fo that in any other 
 cord, by its length. 
 
 PROP. XXIV. 
 
 If a pendulum vibrates in the fmall arch of a cir- j 2 . 
 cle i the time of one vibration^ is to the time of a 
 hodfs falling perpendicularly thro' half the length of 
 the pendulum \ as the circumference of a circle^ to the 
 diameter. 
 
 If a pendulum fufpended by a thread, &c. be 
 made to vibrate in any curve ; it is the fame thing 
 as if it defccnded down a fmooth polifhcd body 
 made in the form of that curve. For the motions, 
 velocities, and times of moving will be the fame 
 in both. 
 
 Let OD or OE be the pendulum vibrating in the 
 arch ADC, whole radius is OD. Let OD be perp. 
 to the horizon, and take the arch E^ infinitely 
 fmall, and draw ABC, EFG, efg^ perp. to OD ; 
 and draw the cord AD. About BD defcribe the 
 femicircle BGD. Draw er and Qs perp. to EG. 
 
 Put / 1= time of defcending; thro' the diameter 
 2OD, or thro' the cord AD. Then the velocities 
 
 gained
 
 30 PENDULUMS. 
 
 Fie. gained by falling thro' 2OD, and by the pendu- 
 
 ,2. lum's delcending thro' the arch AE, will be as 
 
 y/iOD and v^BF. And the fpace defcribed in 
 
 the time /, after the fall thro' 2OD, is 40i), But 
 
 the times are as the fpaces, divided by the vcloci- 
 
 4OO . 
 
 ties. Therefore, . or 2v 2OD : ( (time of 
 
 \/20D 
 
 E^ , 
 
 its defcription) : : .r>r - : time of defcribing Ee zr 
 
 /> ^ Eg 
 
 But by the limilar triangles OEF, Eer ; and 
 
 EF 
 KGF, Ggs-, we Iliall have ^ X Ee — er — F/ 
 
 FG OD X EG 
 
 z= Gj zi jTjy X G^. Whence Ee - ^^ ^ ^p X 
 
 Gg. Therefore the time of defcribing E^ =. 
 / X OD X FG X G^^ __ 
 
 2KD X EF \/W~xlxri5 ~ 
 
 / X QD X\/BF xb DxGg 
 
 2KD \/BF X n/DU + OF X FD X \/aOD 
 /Xv/ODxG^^ _ / X v/^20D' 
 
 2KDXV/DO + OFXx/2 4KDxv'DO+OF 
 / X V/2OD 
 
 in its mean quantity tor all the arches Gg^ is near- 
 ly equal to DK. Therefore the time of defcribing 
 
 Ee n ■^■^= X Gp-. Whence the time 
 
 2BD v/20D — DK 
 
 of defcribinc the arch AED =: _r rrT 
 
 2BD\/20D — DK 
 X BGD. And the time of defcribincr the wlu>le 
 arch ADC, or the time of one ofciUation is n 
 
 /X
 
 Sea. 11. PENDULUMS. 31 
 
 / X v/aOD ___ ^ , , , Fig. 
 
 X 2BGD. But when the arcli 
 
 2BDV/2OD — L>K "^^ ' --'12. 
 
 ADC is exceeding fmall, DK vanilhes, and then the 
 time of olcillation in a very Ihiall arch is =: 
 
 X 2BGD zz -' / X — — — • But if / 
 
 2BD \/zUD ' ^^ 
 
 be the time of defcending thro' 2OD, -^- / is the 
 time of defcending thro' 4- OD. And therefore 
 BD the diameter, \5 to 2BGD the circumference; 
 as the time of falling thro' half the length of the 
 pendulum, to the time of one vibration. 
 
 Cor. I. In a fmall arch AED, the time of de* 
 fcending thro' the cord AD, is to the time of defend- 
 ing thro' the arch AED ; cs the diameter BD, to 
 ~ the circumference. 
 
 For the time of defcending thro' the arch AED 
 
 BGD 
 
 = -^ X —KT^ ; therefore BD : 4- BGD : : / : time 
 zBD 
 
 in AED. 
 
 Cor. 2. All the vibrations of the fame ■pendulum^ in 
 arches not very large, are performed nearly in the fame 
 time. 
 
 Cor. ^. If KD he hiffe5ied in L, and T be :=i 
 time of vibration in a very fmall arch. "Then T + 
 
 KL 
 
 ^^. OK X T will be the time of vibration in any 
 
 arch ADC, nearly. 
 
 For we found tlie time of vibration in ADC — 
 / X BGD 2QD _ 2OD 
 
 BD ^^20D — dK -^^^OD 4-OK ' 
 and the three lines DO + OK, DO 4- OL, and 
 DO + OD are in arithmetical progrefTion •, but 
 flnceKD is very fmall, they are nearly in geometri- 
 cal
 
 32 PENDULUMS. 
 
 Y\cr 2OD DO+OL 
 
 1 2" ^^^ progrefTion -, whence n/^qT OK ~ DO+"oK' 
 
 DO + OL 
 Therefore the tmie of vibration n: T X j^o'IToK 
 
 DO + OK 4 - KL KL 
 
 ^'TX DO 4- OK - ^ "^ ^ ^ DO + OK* 
 
 Cor. 4. Hence a falling body will defcend thro* a 
 fpace of 1 6 feet^ and i inch^ in a Jecond of time. 
 
 For by obfervation, a pendulum 39.13 inches 
 
 BCD 
 long will fwing feconds. And / x "ITiT" = i fe- 
 
 BD 2 
 
 cond, and ^qIj = ^ of 77Z76 ~ ^^"^^ falling 
 
 4 
 
 thro' 2 X 39.13. "Whence (Prop. XIII.) r — 7 ^ : 
 
 3.1416 
 
 39.13 ^ 
 
 2 X 39.13 ••:!': *'*^ X 3.i4i<^ = i93 -09^ 
 
 inches = 16.09 feet. 
 
 PROP. XXV. 
 
 The lengths of two -pendulums., defcrihing fimilar 
 archesy are as the fquares of the t mes of vibration* 
 
 For (Prop. XXII.) the times of defcending thro* 
 two fimilar curves, are as the fquare roots of the 
 lengths of the curves ; that is, as the fquare roots 
 of the lengths of the pendulum, their centers be- 
 ing alike fituated. Therefore the lengths of the 
 pendulums are as the fquares of the times of vi- 
 brating. 
 
 Cor. I. The times of vibration of pendulums in 
 fmall arches of circles.^ are as the fquare roots of the 
 lengths of the pendulums. 
 
 For if the arches arc fimilar, the times of vi- 
 bration are in that proportion. And (Prop. XXIV. 
 
 Cor.
 
 Sea. II. PENDULUMS. 33 
 
 Cor. 2.) if the arches are fmall, tho' not fimilar, Fig. 
 the vibrations will be the fame as before. 
 
 Cor. 2. The velocity of a pendulum at its lowejl 
 pointy is as the cord of the arch it defcends thro'. 
 
 For the velocity at the loweft point is equal to 
 the velocity gained in defcending thro' the cord ; 
 for they are both of them the fame as a body acquires 
 by falling thro* their common altitude. And 
 (Prop. XXIII. Cor. 2.) the velocity gained in 
 falling thro' the cord, is as the length of the cord. 
 Therefore the velocities of a pendulum in different 
 arches, are in the fame ratio. 
 
 PROP. XXVI. 
 
 Pendulumi of the fame length 'vibrate in the fame 
 time, whether they be heai'ier or lighter. 
 
 For let the two pendulums P, p, be of the 
 fame length ; they \n\\ each of theifi fall thro' half 
 the length of the pendulum in the fame time. 
 For (Cor. 2. Prop. V.) the velocity generated in 
 any given time, is as the force dire£tly and mat- 
 ter reciprocally. But in the two pendulums, the 
 forces that generate their motions, are their weights, 
 which are as their quantities of matter. Whence 
 we have the velocity of P, to the velocity of /> ; as 
 
 P p 
 
 ^ to —•> or as I to I : and therefore equal veloci- 
 r p i 
 
 ties are generated in the fame time. Confequently, 
 
 equal fpaces will be defcribed in the fame time, and 
 
 therefore they will fall thro' half the length of one 
 
 of them in an equal time. And therefore (Prop. 
 
 XXIV.) their vibrations v/ili be performed in the 
 
 fame time. 
 
 Cor. Hence all bodies whether greater or kffer^ 
 
 heavier or light rr, near the earth's furfoce will fall 
 
 D ' ' thro"-
 
 34 
 
 PENDULUMS. 
 
 Fio-. thro* equal /paces in equal times j abatifjg the reftjlance 
 of the air. 
 
 Becaute they are as much retarded by their mat- 
 ter, as accelerated by their weight. The weight 
 and the matter being exadtly proportional to one 
 another. 
 
 PROP. XXVII. 
 
 'The lengths of pendulum'' s vibrating in the fame time, 
 in different places of the world j "joill be as the forces 
 of gravity. 
 
 For (by Prop. V. Cor. 2.) the velocity generated 
 in any time is as the force of gravity dired:ly, and 
 the quantity of matter reciprocally. And the mat- 
 ter being fuppoled the fame in both pendulums, 
 the velocity is as the force of gravity •, and the 
 fpace pafled thro' in a given time, will be as the 
 velocity ; that is, as the gravity. Therefore if 
 any two fpaces be defcended thro' in any time, and 
 two pendulums be made, whofe lengths are double 
 thefe fpaces ; thefe pendulums (by Prop. XXIV.) 
 will vibrate in equal times ; therefore the lengths 
 of the pendulums, being as the fpaces fallen thro* 
 in equal timcsj will be as the forces of gravity. 
 
 Cor. I. The times 'ivherein pendulums of the fame 
 length will vibrate, by different forces of gravity \ 
 are reciprocally as the fquare roots of the forces. 
 
 For (Cor. 2. Prop. V.) when the matter is given> 
 the velocity gcfnerated is as the force X by the time. 
 And (Prop. VI.) the fpace defcended thro' by any 
 force, is as the force and fquare of the time. Let 
 thefe fpaces be the lengths of the pendulums, then 
 the lengths of the pendulums are as the forces and 
 the fquarcs of the times of falling thro' them. 
 But CProp. XXIV.) the times of falling thro' them 
 are in a given ratio to the times of vibration ■, 
 
 whence
 
 Sea. II. PENDULUMS. 35 
 
 whence the lengths of pendulums are as the forces Fig. 
 and the fquares of the times of vibration ; there- 
 fore when the lengths are given, the forces will be 
 reciprocally as the fquares of the times ; and the 
 times of vibration reciprocally as the fquare roots 
 af the forces. 
 
 Cor. 2. The lengths of pendulums in different places^ 
 are as the forces of gravity, and the fquares of the 
 times of vibration. 
 
 This is proved under Cor. 2. Hence, 
 
 Cor. 3. The times wherein pendulums of any length, 
 Perform their ofcillations ; are as the fquare roots of 
 *heir lengths dire£lly, and the fquare roots of the gra- 
 ntating forces reciprocally. 
 
 Cor. 4. The forces of gravity in different placesy 
 ire as the lengths of pendulums direElly, and the 
 "quares of the times of vibration reciprocally, 
 
 PROP. XXVIII. Trob. 
 
 To find the length of a pendulum, that fh all make 
 my number of vibrations in a given time. 
 
 Reduce the given time into feconds, then fay, 
 .s the fquare of the number of vibrations given : 
 o the fquare of this number of feconds : : fo is 
 •9.13 : to the length of the pendulum fought, in 
 nches. 
 
 £x. Suppofe it makes 50 vibrations in a minute, 
 lere a minute is — 60 feconds j then. 
 
 As 2500 (the fquare of 50) : 3600 (the fquare 
 
 3600 X 39. 13 
 .f 60) : : 39.13 : to the length — ^^;^^ 
 
 140868 , . , 
 
 "■ ICQ — -- 5^*34 ^"ches, the length required. 
 
 "^ D 2 If
 
 5^ PENDULUMS. 
 
 FW. If it be required to iind a pendulum that fliall 
 vibrate fuch a number of times in a minute j you 
 need only divide 140868, by the fquare of the 
 number of vibrations given, and the quotient will 
 be the length of the pendulum. 
 
 This pradlice is deduced from Prop. XXV. for 
 let p be the length of the pendulum, n the num- 
 ber of vibrations, / the time they are to be per- 
 
 P 
 formed in. Then ^9.13 : 1' : : p : =: fquare 
 
 P 
 
 of the time of one vibration, and v/ — 
 
 , . > . 39-13 
 
 time of one vibration ; then if / be divided by 
 
 P . . . . 39-^3 
 
 */ it will orive ;/ ; that is, / </ — —^ ~ n, 
 
 ^ 39.13 *^ ^ P 
 
 whence// x 39-13 = 'i>^P-> ^nd nn : tl : : ■^^.i^ : p. 
 If the pendulum is a thread with a little ball at 
 it, then the diftance between the point of fufpenfion 
 and the center of the ball is efleemed the length of 
 the pendulum. But if the ball be large, fay as the 
 diftance between the point of fufpenfion, and the 
 center of the bail, is to the radius of the ball ; fo 
 the radius of the ball to a third proportional. Set 
 4- of this from the center of the ball downward, 
 gives the center of ofcillniion. Then the whole 
 diftance from the point of fufpenfion to this center 
 of ofcillation, is the true length of the pendulum. 
 13. If the bob of the pendalum be not a whole 
 fphere, ^Dut a thin fcgment of a fphere, as AB, as 
 in moft clocks ; then to find the center of ofcilla- 
 tion, fay as the diftance between the point of fuf- 
 penfion, and the middle of the bob, is to half the 
 breadth of the bob ; fo half the breadth of the 
 bob, to a third proportional. Set one third of 
 this length from the middle of the bob downwards, 
 gives the center of ofcillation. Then the diftance 
 
 between
 
 Sea. II. PENDULUMS. '^^ 
 
 between the centers of fufpenfion and ofcillation, is Fig. 
 the exad length of the penduliun. 
 
 PROP. XXIX. Proh, 
 
 Having the length of a -pendulum given \ to find 
 how many vibrations it Jhall make, in any given time. 
 
 Reduce the time given into feconds, and the 
 pendulum's length into inches ; then fay, as the 
 given length of the pendulum ; to 39.13 : : fo is 
 the fquare of the time given : to the fquare of the 
 number of vibrations, whofe fquare root is the num- 
 ber fought. 
 
 , Example. Suppofe the length of the pendulum 
 is s^-34- iriches, to find how often it will vibrate in 
 a minute. 
 
 I minute — 60 feconds. Then, 56.34 (the 
 
 length of the pendulum) : 39.13 :- : 3600 (the 
 
 fquare of 60) : to the fquare of the number of vi« 
 
 3600 X 39-13 140868 
 
 brations ziz y— — iz ■'— 7-- — =1 2 ^00, and 
 
 50-34 50-34 -^ ' 
 
 v^2 5oo zz 50 — the number of vibrations fought. 
 
 If the time given be a minute, you need only 
 divide 140868 by the length, and extradb the root 
 of the quotient for the number of vibrations. 
 
 This is the reverfe of the lad problem, therefore 
 fuppofing as before in that problem, we have // X 
 39.13 zz nnp ; therefore^ : 39 13 : : // : «;z. 
 
 They that would fee a further account of the 
 motions of bodies upon inclined planes, the vibra- 
 tions of pendulums, and the motion of proje<5i:iles ; 
 may confult my large book of Mechanics, where 
 they will meet with full fatisfadion. 
 
 D 3 SECT.
 
 
 [38 ] 
 
 Fig- 
 
 
 
 SECT. 
 
 IIL 
 
 Of the Ce7tter of Gravity ; the equtli^ 
 brium of beams of timber ; the di- 
 reBions and quantities of the forces 
 necejjary to fuftain them, 
 
 PROP. XXX. 
 
 A body cannot defcend or fall downijoards^ except 
 enly when it is in Juch a pofttion, that by its motion^ 
 its center of gravity defends. 
 
 \A-' Let the body A ftand upon the horizontal plane 
 BK, and let C be its center of gravity •, draw CD 
 perpendicular to the plane BK. And let the body 
 be fuipended at the point C, upon the perpendicu- 
 lar line CD. Then (def 12.) it will remain un- 
 moved upon the line CD. And as CD is perp. 
 to the horizon, it has no inclination to move one 
 way more than another, therefore it will move no 
 way but remain at reft. Take away the line CD, 
 and let the body be fupported by the line BC •, it 
 BC be fixed, the body will remain at reft on the 
 line CB. But if CB be movable about B, the bo- 
 dy iiifpended at C, will endeavour to move with 
 its center of gravity downwards along the arch CE, 
 about B as a center, towards N. And for the fame 
 reafon the body will endeavour to fall the contra- 
 !y way, moving about the point N j I fay, this will 
 be the cafe when CD is fituated between B and N. 
 But thefe two motions being contrary to one ano- 
 ther, will hi«der each other's effedls j and the body 
 will be fuftained without falling. 
 
 Again,
 
 Sed. IIIv CENTER OF GRAVITY. ^ 
 
 Again, let the body F be fufpended with its cen- Fk 
 ter of gravity I upon the perpendicular IH. As 14 
 this line has no inclination to move to any fide, it 
 will therefore remain at reft. Take away the line 
 IH, and let the center of gravity I be fufpended 
 on the line IG, then the body will endeavour to 
 defcend along the arch IK, for the higheft point of 
 the arch is in the perpendicular ereifted at G. For 
 the fame reafon if the body be fufpended on the 
 line 01, it will endeavour to defcend towards K, 
 about the center O; now as both thefe motions 
 tend the fame way, and there is nothing to oppofe 
 them •, the body muft fall towards K. In both 
 thefe cafes it is plain, that when the center of 
 gravity by its motion, defcends, the body will fall ; 
 but if not, the body will be fupported without falling. 
 
 Cor. I. If a body Jiands upon a plane ^ if a line be 
 drawn from the center of gravity ■perpendicidar to the 
 horizon \ if this line fall within the bafe on which the 
 body flands^ it will be fupported without falling. But 
 if the perpendicular falls without the bafe, the body 
 ivill fall. 
 
 For when the perpendicular falls within the bafe, 
 the body can be moved no manner of way, but the 
 center of gravity will rife. And when t\\t perpen- 
 dicular falls without the bale, towards any fide ; 
 if the body be moved towards that fide, the center 
 of gravity defcends •, and therefore the body will 
 fall that way. 
 
 Cor. 2. If a perpendicular drawn from the center 
 of graxity perp. to the horizon^ fall upon the extremi- 
 ty of the bafe ; the body may ft and., hut the leafi force 
 what ever .^ will caufe it to fall that way. And the 
 nearer the perp. is to any fide., the eofier it will be 
 made to fall., or is fooner thrufl over. And the 
 nearer the perp. is to the middle of the bafe., the firmer 
 the body ftands. 
 
 D 4 Cor,
 
 40 CENTER OF GRAVITY. 
 
 Fig. Cor. 3. Hence if the center of gravity of a body he 
 
 14. fupported^ the whole body is fupported. And the place 
 
 of the center of gravity mufi be deemed the place of 
 
 the body ; and is always in a line drawn perpendicular 
 
 to the horizon, thro'' the center of gravity. 
 
 Cor. 4. Hence all the natural anions of animals 
 may be accounted for from the propertios of the center 
 of gravity. 
 
 When a rnnn endeavours to walk, he ftretches 
 out his hind leg, and bends the knee of his fore 
 leg, by which means his body is thruft forward, 
 and the center of gravity of his body is moved for- 
 ward beyond his feet ; then to prevent his failing, 
 he immediately takes up his hind foot, and places 
 it forward beyond the center of gravity •, then he 
 thrufts himfelf forward, by his leg which now is 
 the hindmoft, till his center of gravity be beyond 
 his fore foot, and then he fets his hind foot for- 
 ward aofain •, and thus he continues walking; as lone; 
 as he pleafes. 
 
 In itanding, a man having his feet clofe toge- 
 ther cannot ftand fo firmly, as when they are at 
 fome diftance ; for the greater the bale, the firmer 
 the body will fland ; therefore a globe is eafily mov- 
 ed upon a plane, and a needle cannot ftand upon 
 its point, any otherwife than by fticking it into the 
 plane. 
 
 When a man is feated in a chair, he cannot rife 
 till he thrufts his body forward, and draws his feet 
 backward, till the center of gravity of his body be 
 before his feet ; or at leaft upon them •, and to 
 prevent falling forward, .he fets one of his feet for- 
 ward, and then he can ftand, or ftep forward as 
 he pleafes. 
 
 All other animals v/alk by the fame rules ; firft 
 fetting one foot forward, that way the center of 
 gravity leans, and then another. 
 
 In
 
 Seft. III. CENTER OF GRAVITY, 41 
 
 In walking up hill, a man bends his body for- Fig. 
 ward, that the center of gravity may lie forward of 
 his feet ; and by that means he prevents his fall- 
 ing backwards. 
 
 In carrying a burthen, a man always leans the 
 contrary way that the burthen lies •, fo that the 
 center of gravity of the whole of his body and 
 the burthen, may fall upon his feet. 
 
 A fowl going over an obftacle, thrufls his head 
 forward, by that means moving the center of gra- 
 vity of his whole body forwards -, fo that by 
 fetting one foot upon the obftacle, he can the more 
 eafily get over it. 
 
 PROP. XXXI. 
 
 hi any tvoo bodies A, B, the common center of gra~ i^. 
 vityQ^ divides the line joining their centers, into two 
 parts, which are reciprocally as the bodies. AC : BC : : 
 B : A. 
 
 ,Let the line ACB be fuppofed an inflexible le- 
 ver ; and let the lever and bodies be fulpended on 
 the point C. Then let the bodies be made to vi- 
 brate about the immovable point C ; then will A 
 and B defcribe two arches of circles about the cen- 
 ter C, and thefe arches will be as the velocities of 
 the bodies, and thefe arches are aifo as the radii of 
 the circles AC and BC. Therefore their velocities 
 are as the radii. Whence, velocity of A : velocity 
 of B : : AC : CB : : (by fuppofition) B : A. There- 
 fore A X velocity of A =z B X velocity of B. 
 Whence fProp. II.) the^ quantities of motion of 
 the bodies A and B are equal, and (Ax. 9.) there- 
 fore they cannot move one another, but muft re- 
 main at reft-, and confequently (def. 12.) C is the 
 center of gravity of A and B- 
 
 Cor.
 
 42 CENTER OF GRAVITY. 
 
 Pig. Cor. I . The produ5Is of each body multiplied by its 
 J r dijiance from the common center of gravity^ are equal. 
 CAXA=:CBXB. 
 
 Cor. 2. If a weight he laid upon C, a point of the 
 inflexible lever AB, which is fupported at A and B ; 
 the preffure at A to the pre£ure at B, will be as CB 
 toCK. 
 
 For let the bodies A, B, be both placed in C ; 
 then (Cor. 3. Prop. XXX.) fince it is the fame 
 thing whether the bodies be at A and B, or both 
 of them at C, their center of gravity •, therefore the 
 prefTures at A and B will be the fame in both cafes. 
 But when they are at A and B, upon the lever 
 ACB, their prelTures aie A and B, being the fame 
 with the weights j therefore when they are both at 
 C, the preflures at A and B will ftill be A and B. 
 Therefore (Cor. i.) fmce it is CA X A zi CB X 
 B i therefore CA : CB : : B : A : : preffure at B ; 
 preffure at A. 
 
 PROP. XXXII. 
 
 15. If there be three or more bodies^ and if a line b^ 
 drawn from one body E to the center of gravity of 
 the refl; C. Then the center of gravity of all the bo- 
 dies divides the line CD, in two parts in D, which 
 are reciprocally as the bcdy E to the fum of all the 
 tther bodies. CD : DE : : E : A + B ^^r. 
 
 ■ For fuppofe AB and CE to be two inflexible 
 lines ; and let the body W — A + B &c. and let 
 W be placed in the center of gravity C. Then 
 by the laft Prop. CD : DE : : E : W or A + B&c. 
 
 Cor. The body E X DE the diflance from the com- 
 mon center of gravity^ is equal to the fum of the bo- 
 dies A + B i^c. X by DC the diflance of their ce»~ 
 Ur from the common center of gravifv. 
 
 PROP.
 
 Sea. in. CENTER OF GRAVITY. 43 
 
 Fig. 
 PROP. XXXIII. 
 
 If A^ B, he two bodies t C their center of gravity, 16. 
 F any point in the line AB. Then will FA X A + 
 
 FB X B = FC X A + B. 
 
 For (Cor. i. Prop. XXXI. J CA X A = CB X B ; 
 that is, FA^^^TfC X A iz FC— -FB X B -, whence, 
 by tranfpofition FA X A + FB X B =:FC XA+B, 
 
 Cor. Hence the bodies A and B have the fame force 
 to turn the lever AF about the point F, as if they 
 were both placed in C their center of gravity. 
 
 PROP. XXXIV. 
 
 If fever al bodies A, B, E ^c. be placed on an 1 7. 
 inflexible freight lever ; and if D be their common 
 center of gravity ; and if F be any point in the line 
 AE, t hen FA X A + F B X B -f FE X E tsf^r. =; 
 
 FD X A + B + E ^i:. 
 
 For if A 4- B = W , then F A x A + FB X 
 
 B + FE X E zz FC X A 4-B + FE x E = FC 
 X W -f FE X E zi (Prop. XXXIII.) FD x 
 
 W + E — FD X A + B 4- F, in the three bo- 
 dies A, B, E. And after the fame manner, if 
 there be four bodies, put W — A -f- B + E, and 
 it will be proved the fame way, that the fum of 
 all the produfts, FA X A + FB x B &c. = dif- 
 tance of the common center of gravity X by all the 
 four. And fo on for more bodies. 
 
 Cor. The fame Prop, will hold good^ when the bo- 
 dies are not in the line AF, but any where in the per- 
 pendiculars paffin? thro" the points A, B, E £ifr. 
 
 PROP.
 
 44 CENTER OF GRAVITY. 
 
 '^* PRO P. XXXV. 
 
 17- If there he any number of bodies A, B, E, ^c. 
 either placed in the line AF, or any ivay in the per- 
 pendiculars pi'fjmg thro' A, B, E. And if D be 
 the centey of gravity of all the bodies ; and F be any 
 point in the line AF. Then the difiance of the center 
 
 . ^^ FAXA + FBXB + FEXE 
 of gravity FD - a + B + E 
 
 For whether the bodies be in the points A, B, 
 E, or in the perpendiculars, it will be (by Prop. 
 XXXIV. and Cor.) that FA X A + FB X B + 
 
 FE X E =: FD X A + B + E. Whence FD = 
 A X FA + B X FB + E X FE . . „ , 
 
 A4.B4-E — ~7 = ^""^ ^^ "" '^" 
 
 produds of each body multiplied by its diftance, 
 divided by the fum of the bodies. 
 
 Cor. 1. If a fingle body only was placed on the k- 
 Ter AF ; then the diftance of the center of gravity of 
 that hody^ is equal to the fum of the produ^s of all 
 the particles of the body^ each multiplied by its dif- 
 tance from a given point F, and divided by the body. 
 
 For if A, B, E &c. are feveral particles of tlie 
 body, then A + B -f E &c. — the body ; and 
 
 _ A X FA + B X FB + E X FE 
 ^^ - body 
 
 Cor. 2. If there be feveral bodies A, B, E, i^c^ 
 placed upon the lever AF. They a£l with the fame 
 force in turning the lever about any given point F, as 
 if they were all placed in D the common center of gra^ 
 vity of all the bodies. 
 
 Scholium. 
 
 If any of the bodies be placed on the contrary fide 
 of F, their refpe^Hve produ^s will be negative. 
 
 For
 
 Sea. III. CENTER OF GRAVITY. 45 
 
 For they a6t the contrary way in turning the le- Fig. 
 ver about. 17. 
 
 PROP. XXXVI. 
 
 ■ If fever al bodies A, B, E, G, H, he placed or' 18. 
 the lever AH, mid F be the center of gravity of all 
 the weights. "Then FA X A + FB X B 4- FE X E 
 = FG xG + FHxH. 
 
 For let the lever be fufpended on the point F, 
 then the two ends will be in equilibrio, as F is the 
 center of gravity. Let D be the center of gravi- 
 ty of A, B, E ; and I the center of gravity of G, 
 H. Then (Cor. Prop. XXX III.) it is the fame 
 thing whether the bodies on one fide be placed at 
 A, B, E, or all of them in the point D. And 
 whether thofe at the other end be placed at G, H; 
 or all of them at I. But fince F is the center of 
 
 gravity, DF X A + B + E = FI X G + H, and 
 therefore A X AF + B X BF + E X EF zz G X 
 GF + H X HF (by Prop. XXXIVJ 
 
 Cor. I. If fever al bodies A, B, E, G, H, he 
 
 placed on an inflexible lever., and if AX FA ~j- B X 
 FB + E X FE zz G X FG + H X FH. 1'bin F 
 
 is the center of gravity of all the bodies. 
 
 For no other point will aniwer the equation. 
 
 Cor. 2. If fever al bodies A, B, E, G, H, be 
 placed upon a lever AH, or fufpended at thefe points 
 by ropes ; and if Ax FA + B X FB -f E X FE zz 
 G X EG + H X FH i they ivill be in equilibrio upon 
 the point F. 
 
 This appears by Def. 12, and F is the center of 
 gravity. 
 
 PROP.
 
 46 CENTER OF GRAVITY. 
 
 Fig. 
 
 PROP. XXXVII. 
 
 J 9" If a hea-vy body AB, fufpended by two 7' opes AC, 
 BD, remains at reft \ a right line perpendicular to the 
 horizon^ pajjing thro* the interfeBion F of the ropes \ 
 will alfo pafs thro' the center of gravity G, of the body. 
 
 If AC and BD be produced to F where they in- 
 terfedlj then (ax. 12.) it is the fame thing Vv'hether 
 the force ading in diredtion AC be applied to C 
 or F •, and whetlier the force afting in direftion BD 
 be applied to the point D or F. Siippofe then that 
 they both a6l at F, and then it is the fame thing, 
 as if the body was fufpended at F by the two firings 
 AF, BF. And fmce the body is at reft, therefore 
 (Ax. 7.) the body, that is, the center of gravity 
 G, is at the loweft place it can get ; and therefore 
 is in the plumb line FG. For if the body be made 
 to vibrate, the center of gravity G will defcribe an 
 arch of a circle, of which G (being in the perp. 
 FG) is the loweft point. 
 
 Cor. I. He7ice if GN be drawn parallel to AC; 
 then the weight of the body, the forces a^ing at C 
 and D, are refpetUvely as FG, GN, and FN. 
 
 This is evident by Prop. VIII. 
 
 Cor. 2. If a heavy body AB, be fupported by two 
 planes, IKL, and EHG, at H and K ; and HF a7td 
 KF be drawn perpendicular to thefe planes ; and if FG 
 be drawn from the i7it€7fe5fion F, perp. to the hori- 
 zon, it will pafs thro' the center of gravity G, of the 
 body. 
 
 For fince the body is fuftained by thefe planes, 
 therefore the planes re-a6b againft the body (by 
 Prop. IX,), in the diredlions HF, KF pependicu- 
 lar to thefe planes. Therefore it is the fame thing 
 
 as
 
 C 3) 
 
 6. 
 
 tB 
 
 H 
 
 12. 
 
 15- 
 
 S'^rf 
 
 PlX/'A^<f.
 
 TlL/ra-jlf. 
 
 c>
 
 Sea. III. CENTER OF GRAVITY. 47 
 
 as if the body was fuftained by the two ropes HP, Fig. 
 KF. For the diredions and quantities of the 19. 
 forces, ading at H and K are the fame in both 
 cafes. And further, if the body be made to vi- 
 brate round F, the points H, K will defcribe two 
 arches of circles, coinciding with the touching 
 planes at H, K-, therefore moving the body up 
 and down the planes, will be juft the fame thing, 
 as making it vibrate in the ropes, HF and KF ; 
 and confequently, the body can reft in neither cafe, 
 but when the center of gravity G is in the perpen- 
 dicular FG. 
 
 Scholium. 
 
 If any body fhould deny the truth of this Prop. 
 or its corollaries, againft the cleareft force of de- 
 monftration. It lies upon them to fhew where the 
 deraonftration fails, or what ftep or fteps thereof do 
 not hold good, or are not truly deduced from the 
 foregoing. If they cannot do this, what other rea- 
 fons they may afllgn, can fignify nothing at all to 
 the purpofe. And if fuch perfon, ignorant of the 
 laws of nature, and the refolution of forces, would 
 objedl agairft this praflice, of fubftituting planes 
 perpendicular to the lines or cords fuftaining any 
 weights, inftead of thefe cords. Let him firft read 
 Sir J. Neivion^s Principia, Cor. 2. to the laws of 
 nature, where he will fee this praftice exemplified, 
 and then make his objecftions. 
 
 And for the fake of fuch perfons as underftand 
 not how to apply the method of compofition and 
 refolution of forces, I fhall add a few problems to 
 prevent their being milled by the rafh judgment of 
 fome people, who having brought out falfe folu- 
 tions to fome problems by their own ill manage- 
 ment, condemn the method as erroneous •, wheii 
 the fault really lies in their own ignorance, and not 
 at all in the method itfelf. 
 
 PROP.
 
 48 CENTER OF GRAVITY. 
 
 ^*g- PROP. XXXVUI. Prob. 
 
 20. <j'q determine the pofition of a beam CD, movable 
 about the end C, and fujlained by a given weight Q, 
 banging at a rofe Q AD, "juhich goes over a pulley at A 
 and is fixed to the other end D. 
 
 Draw AF, CK parallel to the horizon, FDE 
 perp to it, and KL) perp. to CD ; and let G be 
 the center of gravity, w zz Vv^eight of the beam. 
 Then if the beam was to lie horizontally (Prop. 
 XXXI. Cor. 2.) it would be GC : GD : rprefTare 
 at D : preffure at C j and GC : CD : : prelilire at 
 
 GC 
 D : ^ ; whence the prelTure at D zi pttt w, hori- 
 zontally. And (Prop. XIV.j CD : CE : : 
 
 GC CExCG . ,. ^. ^^^ 
 
 p=r w i — VTTTT — iu •=. preffure m diremon DK. 
 
 Produce AD to O, and draw OI parallel to CD. 
 Then the beam is fuftained by three forces in direc- 
 tions OD, DI and lO •, and DI : DO : : S.IOD or 
 CDC or ADC : rad ^ v/hence S.ADC : rad : : 
 CE X CG rad X CE X CG 
 
 — ^r- X w : force DO or Q z= g /^pc X CD^ ^^'• 
 
 CE X CG 
 
 Therefore w : Q^: : S.ADC : rad X — t^.tt — : : 
 
 CG rad x CE 
 S.ADC : ^ X S.FDC, becaufc -^^ — = 
 
 S.EDC or FDC. 
 
 PROP. XXXIX. Prob. 
 2 1. Let the beam ED, be fujlained by the weights P, Q» 
 by means of the rcpes DCP, EACT, going over the 
 pulleys C, A, in the horizontal line AC. To find the 
 pcfition of the beam \ having the weights P, Q, given. 
 I JVay. 
 Let G be the center of gravity of the beam. 
 Thro' D, E, draw HDS, FER perpendicular to 
 
 AC.
 
 Sea. III. CENTER OF GRAVITY. 4^ 
 
 AC; Then (Cor. 2. Prop. VIII.) S.EDS : P the Fio-. 
 tenfion of the thread CD : : S.CDS or CDH : 2T 
 S.CDH ^_ 
 
 g£jpg X P the tenfion of DE. Alfo, S.DER or 
 
 S.AEF 
 EDS : Q^: : S.AER or AEF : -jEm ^ ^ ^^® 
 
 tenfion of DE in a contrary dire6lion. Then as 
 
 the beam is in equilibrio, thefe forces or tenfions 
 
 , , . S.CDH 
 
 balance one another j therefore <. ,~^^. X P =: 
 
 S.AEF 
 
 g-^^ X Q. Then P : Q : : S.AEF : S-CDH; 
 
 which may be otherwife exprefled, for AE : rad : : 
 
 AF 
 AF : S.AEF 1= -jgX rad ; and DC : rad : : HC : 
 
 HC AF 
 
 ^ X rad n S.HDC. Whence P : Q.^ • ^ > 
 
 CH AF AE 
 
 CD-'HC'DC*' 
 
 2 Way. 
 
 Let R, S be the perpendicular preffures of the 
 ends E, D. w = weight of the beam. Then 
 
 dg eg 
 
 (Cor. 2. Prop. XXXI.) R — ^^ w, and S n g^w. 
 
 DG 
 And (Cor. 2. Prop. VIII.) S.AED : R or ^w : ; 
 
 S.AEF X DG 
 S.AEF : tenfion of DE — g ^ED X ED "^^ ^^'^ 
 
 EG ' ^ 
 
 S.CDE : S or gr^ w : : S.CDH : contrary tenfion 
 
 S.CDH X EG , , ^ 
 
 of DE zz c CDF y FD ^' ^ ^^° forces 
 
 S.AEF XDG 
 ©f DE being equal, we have 5 aED X ED "^ ^ 
 
 E ' S.CDH
 
 50 CENTER OF GRAVITY. 
 
 Fig. SC DHxEG S.AEFxDG _ S.CDHxEG 
 
 2u SCDExIED'^'^"'^ S.AED - S.CDE 
 
 S.AEF S.CDH 
 Whence EG : DG : : g^^^jj : g^^DE * '' ^'^^^ 
 X S.CDE : S.CDH X S.AED. 
 
 S FDS 
 S.CDE : S.EDS : : S : P = ^^^ X S, and 
 
 „ S.RED 
 S.AED : S.RED : : R : g"^^ R = Q^ Then P: 
 
 S.EDS S.RED 
 
 ^^ '' -sZm X S '- STaED X R = = S.AED X S : 
 
 EG 
 S.CDE X R : : (laft method) S.AED X jg^- w : 
 
 DG 
 S.CDE X^w:: S.AED X EG : S.CDE X DG. 
 
 And S.AED : S.CDE : : P X DG : Q^X EG. 
 
 4 IFay. 
 
 Draw Cm, Fn parallel to DE, and FE, HD 
 perp. to the horizon. Then by the refolution of 
 
 "Dm 
 forces, CD : D;;; : : P : ^^ P = perpendicular 
 
 EF 
 
 force at D j and nE : EF : : Q^: ~p- Q =z perpen- 
 
 dicular force at E. Therefore EG : GD ; : p^ P : 
 EF S.CDE ^ S.FkE _ ^^^^ 
 
 ^Q-^o;;dxp = s:;;feX^-s-^^^x 
 
 P : S.AED X Q^ ForS.C;;zD zz S.mDE = S.FED 
 zz Sj;FE. That is, EG : GD : : S.CDE X P : 
 S.AED X Q. As in the third way. 
 
 5 PFay. 
 
 T,et R, S be the perpendicular weights of the 
 f nds E, D ', or v/hich is the lame, the tenfions of the 
 
 perpen-
 
 Se(5b. III. CENTER OF GRAVITY. 51 
 
 dicular ropes, FE, HD. By the refolution of Fig» 
 forces, if Cm, ¥n be parallel to DE. The force 21, 
 FE or R is equivalent to E;/, ¥n ; and the force - 
 Dm or S, to DC, mQ ; therefore EF : F» : : R : 
 
 F« 
 
 pp- R ■=. force at E in direflion Vn, And "Dm : 
 
 wC : : S : -^ S z= force at D in direftion mC But 
 
 the beam being in equilibrio, thefe two oppofite 
 
 F?z mC 
 
 forces muft be equal; therefore pp R ^ ""n^* 
 
 c ^ ll S-CDH S.AEF 
 Whence R ^ S : : ^^^-^ : ^^ : : § ^DE * S. AED ' ' 
 
 S.AEDxS.CDH : S.AEF X S.CDE. But (Cor. 2. 
 Prop. XXXI.) R : S : : DG : EG. Whence DG : 
 EG : : S.AED X S.CDH : S.AEF X S.CDE ; the 
 fame as by the 2d method. And the fame thing 
 likewife follows from the ift and 4th method to- 
 gether. 
 
 From thefe feveral ways of proceeding, it is evi- 
 dent, that Vv^hich ever way we take, the procefs if 
 rightly managed alv/ays brings us to the fame con- 
 clufion -, and it comes to the fame thing which way 
 we ufe, fo that we proceed in a proper manner. 
 And this among other things, fhews the great ufe 
 and extent of that noble theory of the compofi- 
 tion and refolution of forces. 
 
 What is calculated above is concerning the an- 
 gles, or the portion of the feveral lines to one another, 
 depending on the feveral forces. Then in regard to 
 the weight of the beam, put it =r w^ then DC : 
 
 Bm EF 
 
 Dw : : P : g^ P =^ S, and E« : EF : : Q^: ^ Q 
 
 Vim EF 
 
 = R. Andw = R-{-S=|^P + £^Q, an 
 
 equation fhewing the relation of the weights to 
 one another. 
 
 E 2 6 fVayj
 
 52 CENTER OF GRAVITY. 
 
 2 f, 6 Way^ hy the center of gravity. 
 
 Produce AE* CD to B, and from B draw BGO 
 thro' the center'of gravity; which (by XXXVll.) 
 will be perp. to AC, and therefore parallel to EF, 
 DH. Then the angle EBG — AEF, and DBG 
 — CDH. Then EB : BD : : S.BDE or CDE : 
 S.BED or AED, and (Trigonom. B. II. Prop. V. 
 Cor. I.) EG:GD : : EB XS.EBG : BDX S.DBG:: 
 EB X S. AEF : BD X S.CDH : : S.CDE X S.AEF : 
 S.AED X S.CDH ; the fame as by the 2d way. 
 Whence all the reft will be had as before. 
 
 Cor. // will he exactly the fame things whether 
 the weights P, Q, remain^ or the firings AE, CD, 
 be fixed in that pofition to two tacks^ any way in 
 thefe lines. And if a beam ED, hang upon two 
 tacks A, C, by ropes fixed there ; // makes no dif- 
 ference^ if you put two pulleys infiead of the tacks^ 
 for the ropes to go over^ and then hang on the weights 
 Q, P, efiual to the ten/tons of the firings AE, CD. 
 
 For in both cafes, the forces or the tenfions of 
 the ftrings, and their dire6lions, remain the fame. 
 And there is nothing elfe to make a difference in 
 the fituation of the beam. 
 
 Scholium. 
 Ever)'' one that knows any thing of mechanical 
 principles will eafily underftand, that if any forces, 
 which keep a body at reft, be refolved into others, 
 to have the fame effeft ; the contrary forces, or 
 ■" thofe direftly oppofitc, muft aft againft a fingle 
 point ; or elfe the equilibrium will be deftroyed. 
 And therefore in the prefent Prop, fuppofe any one 
 fticuld divide the forces CD, AE, into the two HD, 
 DY, and FE, EX, one perpendicular, the other 
 parallel to the horizon. The forces HD, EF, will 
 indeed balance the force of gravity at D and E, to 
 
 which
 
 Sea. III. CENTER OF GRAVITY. S3 
 
 which they are direftly oppofite. And therefore Fig. 
 the beam will remain unmoved by thefe. But the 2;. 
 equal forces DY, EX, being parallel, never meet in 
 a point •, but ading obliquely on the beam, one of 
 them drawing at D in direction DY, and the other 
 at E in diredtion EX, the effed will be, that they 
 will turn the beam ED about the center of gravi- 
 ty G. Therefore to prevent this, the forces DY, 
 EX, muft be fubdivided ; that is, they muft be 
 refolved into others, one whereof is perp. to the 
 horizon, the other parallel to ED. Then gravity 
 will balance thefe perp. to the horizon, and the 
 others, being equal and oppofite, afting in the line 
 EGD, adl equally againft any of the points D, G, 
 or E. And lb the beam will remain at reft. But 
 this is much better done at once at the firft, by di- 
 viding DC, EF, each into two forces, one perp. to 
 the horizon, the other parallel to the beam ED. 
 And then the oppofite forces will exactly balance 
 one another, and the beam remain unmxoved. 
 
 PROP. XL. Pro]>. 
 
 To find the pojltion of the beam ED, hanging by 2 2-. 
 the rope EBD, whofe ends are fafiened at E and D, 
 and goes over a pilley fixed at B. 
 
 Let G be the center of gravity of the beam, then 
 
 (Prop. XXXVII.) BG will be perp. to the horizon. 
 
 Then as the cord runs freely about the pulley B ; 
 
 therefore (Ax. 13.) the tenfion of the parts of the 
 
 rope EB, BD are equal to one another, fuppofe 
 
 :z: T. By the refolution of forces, the force EB 
 
 is equivalent to EG, GB •, and DB to DG, GB. 
 
 EG 
 Therefore BE ; EG : : T : TfoT zz force m direc- 
 
 DG 
 tion EG, And BD : DG : : T : gg T - force in 
 
 E 3 diredtion
 
 54 CENTER OF GRAVITY. 
 
 Fig. direflion DG, which is equal and oppofite to that 
 
 22 r E^ rr^ ^^ 
 
 in EG J therefore ^^T — ^g T. Whence EG: 
 
 EB : : DG : DB. And therefore BG bifeds the 
 angle EBD. 
 
 Cor. Hence ED : firing EBD : : EG : EB the part 
 EB of the firing : : and Jo GD : DB ihe fart DB of 
 the firing. 
 
 Scholium. 
 
 If GD be lefs than GE, then the center of gra- 
 vity G, will be loweft, when the beam hangs per- 
 pendicular with the end D downward. And in ma- 
 ny cafes it will be higheft, when it hangs perpen- 
 dicular, with the end E downward. 
 
 PROP. XLI. Proh. 
 
 20, 'There is a beam BC hanging by a pin at C, and 
 lying upon the is: all BE •, to find the forces or preffures 
 at the points B, and C, and their directions. 
 
 Produce BC to K, fo that CK may be equal to 
 CB. Draw BA parallel, and CL perpendicular, to 
 the horizon ; and draw BL, CN, KI perp. to 
 BCK. Thro' the center of gravity G, draw GF 
 parallel to CL. By Prop. XIV. if a body lies 
 upon an inclined plane, as BC ; its weight, its in- 
 clination down the plane, and prefllire againft it, 
 are as BC, CA and AB ; that is, as CL, CB and 
 BL. Therefore if CL reprefent the weight of the 
 body, CB will be the force urging it down the 
 plane, and BL the total prefTure againft the plane. 
 And fincc GF is parallel to CL, BL is divided in 
 F, in the fume ratio, as BC is divided in G. And 
 therefore (Cor. 2. Prop. XXXI.) BF will be the 
 part of the prcflTure afting at C, and FL the part 
 afting at B. Make CN equal to BF, and compleat 
 
 the
 
 Scdl. III. CENTER OF GRAVITY. 55 
 
 the parallelogram CNIK, and draw CI. Then Fig, 
 fince BC or CK is the force in diredtion CK, and 23. 
 CN the force in diredion CN ; then by compofi- 
 tion, CI will be the fingle force by which C is fuf- 
 tained, and CI its direftion. But tjfie triangles CKI, 
 CBF are fimilar and equal, and CI ::z: CF, and in 
 the fame right line ; therefore CF is the quantity 
 and direction of the force afting at C to fuftain it. 
 Therefore the weight of the body, the prefllire at 
 B, and the force at C j are refpeftively as CL, FL, 
 and CF. 
 
 Cor. I . Produce FG to interfe5f CN in H ; then 
 the weight of the body, the prejfure at B, and the 
 force aSiing at C ; are refpeSiively as HF, HC, and 
 CF. 
 
 For in the parallelogram CLFFI, HF z= CL, 
 and HC = FL. 
 
 Cor. 2. If the beam was fupported by a pin at B, 
 and laid upon the wall AC -, the like confiru5lion mufi 
 he made at B, as has been done at C, and then the 
 forces and directions will be had. 
 
 Cor. ^. If a line per p. to the horizon be drawn 
 thro* F, where the direkion of the forces CF, and 
 BF meet ; it pajjes thro' G the center of gravity of 
 the beam. 
 
 Cor. 4, // is all one whether the beam is: fufiained 
 by the pin C and the wall BE, or by two ropes CI, 
 BP a5ling in the dire^ions FC, FB, and with the 
 forces CF, FL. 
 
 Scholium, ' 
 
 The proportions and direftions of the forces here 
 found, are the fame as in Prop. LXIV. of my large 
 book of Mechanics, and obtained here by a dif- 
 ferent method. The principles here ufed are in- 
 diiputable ; and the principle made ufe of in that 
 E 4 LXiV.
 
 56 CENTER OF GRAVITY. 
 
 Fig. LXIV. Prop, is here demonftrated in the third Cor. 
 
 23. fo that the reader may depend upon the truth of 
 them all. 
 
 PROP. XLII. Proh. 
 
 24. BC is a heavy beam fupported upon two pojls KB, 
 LC •, to find the pofuion of the poftsy that the beam 
 ma) refi in e^uilibno. 
 
 Let G be the center of gravity ; draw BA paral- 
 lel to the horizon, and BF, GD, CAN perpendi- 
 cular to it. Then (Prop. XXXI. Cor. 2.) if BC be 
 the weight of the body, CG will be the part of 
 the weight afling at B, and BG the weight at C. 
 Therefore make CN — BG, and BF zz GC i and 
 from N and F, draw NI, FK, parallel to BC ; and . 
 make NI zz. FK, of any length, and lying con- 
 trary ways. Then draw IC and KB, which will be 
 the pofition of the pofts required. 
 
 For BF is the weight upon B ; and CN, that 
 upon C, which forces being in diredion of the lines ■ 
 BF, CN, the beam will remain at reft by thefe 
 forces. _, And the forces NI, FK, in diredlion BC, 
 being equal and contrary, will alfo keep the beam in 
 equilibrio, therefore the forces KB, IC, compound-^ 
 ed of the others, will alfo keep the beam in equi- 
 librio, ading in the diredions KB, IC, or MB, LC. 
 
 Cor. I . Hence if DG be produced, it willpafs thro* 
 the interfe5iion H, of the lines LC, MB. 
 
 For the triangles INC, CGH are fmiilar •, there- 
 fore IN : NC : : CG : GH, the interfedlion with 
 CL. Alfo the triangles KFB, BGH are fimilar; 
 therefore KF : BG : : BF : GH the intcrfedion 
 with MB, which mull needs be the fame as the 
 other, fmce the three firft terms of the proportion 
 are the fame j for KF — NI, BG — NC, and 
 BF - CG. 
 
 Cor,
 
 Sea. III. CENTER OF GRAVITY. c^-j 
 
 Cor. 2. If a line be drawn thro* the center of gra- Fig. 
 'uity G, of the beam^ perpendicular to the horizon-, 24. 
 and from any point H in that line, (above or below 
 G), the lines HBK and HCM be drawn-, then the 
 props BM and CL will ftiflain the beam in equilibrio. 
 
 Cor. 3. If GO be drawn parallel to HC ; then the 
 weight of the beam, the prefftire at C, and thrufi or 
 preffure at B ; are refpe£lively as HG, OG, and HO, 
 
 and in thefe dire^ions. 
 
 Cor. 4. It is all one for maintaining the equilibri- 
 um, whether the beam BC be fupported by two pofts 
 or props MB, LC ; or by two ropes BH, CH ; or 
 by two planes pe^yendicular to thefe ropes at B and C. 
 
 For in all thefe cafes the forces and diretSlions 
 are the fame •, and there is nothing elfe concerned, 
 but the forces and diredlions. 
 
 Scholium. 
 
 It does not always happen that the center of 
 gravity is at the lowed place it can get, to make 
 an equilibrium. For here if the beam BC be fup- 
 ported by the pofts MB, LC •, the center of gra- 
 vity is at the higheft it can get ; and being in that 
 pofition, it has no inclination to move one way 
 more than another, and therefore it is as truly in 
 equilibrio, as if it was at the lowed point. It is true, 
 any the leaft force will deftroy that equilibrium, 
 and then if the beam and pofts be movable about 
 the angles M, B, C, L, which is all along fuppofed, 
 the beam will defcend till it is below the points M, 
 L, and gain fuch a pofition as defcribed in Prop. 
 XXXIX. and its Cor. fuppoftng the ropes fixed at 
 A, C (fig. 21.) •, and then G will be at the loweft 
 point, and will come to an equilibrium again. In 
 planes, the center of gravity G may be either at its 
 jiigheft or loweft point. And there are cafes, when 
 
 the
 
 5S CENTER OF GRAVITY. 
 
 Y'lcr. the center of gravity is neither at its higheft nor 
 24. lowed, as may happen in the cafe of Prop. XL. 
 fo that they who contend, that in cafe of an equi- 
 librium, the center of gravity mull always be at 
 the loweft place, are much miftaken, and know 
 little about the principles of mechanics, or the na- 
 ture of things. 
 
 Thofe that want to fee more variety about the 
 motion of bodies, on inclined planes •, the preflure, 
 and direftion of the prefTure of beams of timber i 
 centers of gravity, and alfo the centers of ofcilla- 
 tion and percufTion, &c. may confult my large 
 book of Mechanics. 
 
 SECT.
 
 [59] 
 
 Fig 
 
 SECT. IV. 
 
 72^ Mechanical Powers; the Strength 
 and Strefs of Timber » 
 
 PROP. XLIII. 
 
 JN ci balance, ivhere the arms are of equal length ; 
 if two equal weights he fufp ended at the ends^ they 
 will he in equilihrio. 
 
 The balance is a flreight inflexible rod or beam, 
 turning about a fixed point or axle in the middle 
 of it ; to be loaded at each end with weights fuf- 
 pended there. 
 
 Let AB be the beam or lever, C the middle 25. 
 point or center of motion -, D, E the weights 
 hanging at the ends A and B. Then let the beam 
 and the weights, or the whole machine, be fufpend- 
 ed at C. And fuppofe the beam and the weights 
 be turned about upon the center C; then the 
 points A, B being equidiftant from C will defcribe 
 equal arches, and therefore the velocities will be 
 equal, and if the bodies D and E be equal, then 
 the motion of D will be equal to the motion of E, 
 as their quantities of matter and velocities are 
 equal -, and confequently, if the beam and weights 
 are fet at reft, neither of them can move the other, 
 but they will remain in equilibrio. 
 
 Cor. If one weight he greater than the other \ 
 that weight and fcale will defend, and raife the other. 
 
 Scholium. 
 The ufe of the balance, or a common pair oF 
 fcales, is to compare the weights of different bo- 
 dies j
 
 6o MECHANIC POWERS. 
 
 Fig. dies ; for any body whofe weight is required, be- 
 25. ing put into one fcale, and balanced by known 
 weights put into the other fcale, thefe weights will 
 fhew the weight of the body. To have a pair of 
 fcales perfedl, they muft have thefe properties. 
 I. The points of fufpenfion of the fcales, and the 
 center of motion of the beam. A, C, B, muft be 
 in a right line. 2. The arms AC, BC, muft be 
 of equal length from the center. 3. That the cen- 
 ter of gravity be in the center of motion C. 
 
 4. That there be as little fridlion as pofiible. 
 
 5. That they be in equilibrio, when empty. 
 
 If the center of gravity of the beam be above 
 the center of motion, and the fcales be in equili- 
 brio, if they be put a little out of that pofition, 
 by putting down one end of tlie beam, that end 
 will continually defcend, till the motion of the 
 beam is ftopt by the handle H. For by that mo- 
 tion, the center of gravity is continually defcend- 
 ing, according to the nature of it. But if the cen- 
 ter of gravity of the beam be below the center of 
 motion -, if one end of the beam be put down a 
 little, to deftroy the equilibrium -, it will return 
 back and vibrate up and down. For by this motion 
 the center of gravity is endeavouring to defcend. 
 
 PROP. XLIV. ProL 
 
 25. jTo make a falfe balance •, or one which is in equi- 
 librio when empt}\ and alfo in equilibrio, when loaded 
 with unequal weights. 
 
 Make fuch a balance as defcribed in the laft 
 Prop, only inftead of making the center of mo- 
 tion in the middle at C, make it nearer one end, 
 as at F. And make the weight of the fcales fuch, that 
 they may be in equilibrio upon the center F Then 
 if two weights D, E, be made to be in equilibrio 
 in the two fcales ; thefe weights will be unequal, 
 
 for
 
 Sea.lV. MECHANIC POWERS. 6i 
 
 for they will be reciprocally as the lengths of the Fig. 
 armsAF, BF. That is, AF : BF : : E : D. 25. 
 
 For (Prop. XXXI. Cor. i.) fince F is the cen- 
 ter of gravity of D and E, fuppofing them to aft 
 at A and B ; therefore FA X D — FB x E. And 
 FA : FB : : E : D. But AF is greater than FB, 
 therefore E is greater than D. 
 
 Cor. I. Hence to dif cover a falfe balance, make the 
 weights in the two fcales to be in equilibrio j then 
 change the weights to the contrary fcales. And if 
 they be not in equilibrio, the balance is falfe. 
 
 Cor. 2, Hence alfo to 'prove a pair of good fcales, 
 they muft be in equilibrio when empty, and likewife in 
 equilibrio with two weights. Then if the weights be 
 changed to the contrary fcales, the equilibrium will 
 ftill remain, if the fcales are good. 
 
 Cor. 3. From hence alfo may be known what is gain^ 
 td or lofi, by chatiging the weights, in a falfe balance. 
 
 Take any weight as i pound, to be put into 
 one fcale and balanced by any fort of goods in the 
 other. Since AF X D ~ BF X E j let. the weight 
 
 AF 
 D be I, then E zz ^ the weight of the goods in 
 
 the fcale E. Then changing the fcales, let the 
 
 BF 
 
 weight E be I ; then D iz ~r^ the weight of the 
 
 J . 1 r , T. ^, AF BF 
 
 goods in the Icale D. Then ^ + xp zz whole 
 
 • L r 1. J , AF BF 
 
 weight ot the goods, and gp + y:^p — 2 = gain 
 
 to the buyer in 2 lb. &c. Therefore 
 AF^ -V BF^ — 2AF X BF 
 
 ' A F X BF g^^^ ^^ 2 lb. - 
 
 AF — Bt^' AF — BF^' . . 
 
 AFXBF-' '"'^ ^I^FJTbF = ^'-^^ ^" ' ^^- 
 
 There-
 
 e>^ MECHANIC POWERS. 
 
 Fig. Therefore if w is any weight to be bought ; the 
 
 25. gain to the buyer, for the weight w, by changing 
 
 the fcales, will be — -—^ — 17=- w. For example, 
 2Ar X Br 
 
 if AF be 1 6, and BF 1 5 i then the gain will be 
 16—15'' ^ 
 
 2X16X15 400 
 
 Scholium. 
 
 In demonftrating the properties of the mecha- 
 nical powers ; fmce the weight is commonly fome 
 large body whofe weight is to be overcome or ba- 
 lanced •, therefore the power which afts againft it, 
 will be moft fitly reprefented by another weight ; 
 and thus the power and weight being of the lame 
 kind, may moft properly and naturally be com- 
 pared together. For fuch a weight may reprefent 
 any power, as the ftrength of a man's hand, the 
 force of water or wind, &c. And this weight re- 
 prefenting the power, being fufpended by a rope, 
 may hang perpendicular where the power is to ad 
 perpendicular to the horizon ; or may go over a 
 pulley, when it a6ts obliquely. 
 
 PROP. XLV. 
 
 26. ^f i^^ power and weight he in eauilihrio upon any 
 
 27. lever ^ and a^ in lines perpendicular to the lever \ 
 
 28. then the power P is to the weight W ; as the dif~ 
 2Q. tance of the weight from the fiipport C, is to the dif* 
 
 tance of the power from the fupport. 
 
 There are four forts of levers, i. When the 
 fupport is between the v/eight and the power. 
 2. When the wfight is between the power and the 
 fupport. 3. When the power is between the weight 
 and the fupport. 4. When the, lever is not ftreight 
 but crooked. 
 
 Ale-
 
 Sea. IV. MECHANIC POWERS. 63 
 
 A lever is any inflexible rod or beam, of wood Flo-, 
 or metal, made ufe of to move bodies. The fup- 26. 
 port is the prop it reftson, in moving or fuftaining 27. 
 any heavy body, and this is the fame as the center 28. 
 of motion. 29. 
 
 Let the power P a<5t at A by means of a rope ; 
 then as C is the prop or center of motion, if the 
 lever be made to move about the center C, the 
 arches defcribed by A and W •, that is, the veloci- 
 ties of A and W will be as the radii CA and CW. 
 But the velocity of P is the fame as that of the 
 point A. Therefore velocity of P : velocity of 
 W : : CA : CW : : (by fuppofition) W : P ; there- 
 fore P X velocity of P r: W X velocity of W. 
 Confequently their motions are equal, and they 
 cannot move one another, but muft remain in equi- 
 librio. And if they be in equilibrio, they muft 
 have this proportion afligned. 
 
 Cor. 1. Jf a power P a5l ohliq^uely againft the k- 3°' 
 'ver W A •, the 'power and weight 'will he in equilihrio^ 
 when the -power P is to the weight W ; as the dif- 
 tanceof theweightCW^toCD the perpendicular, drawn 
 from the fupport to the line of dire^ion of the power. 
 
 For in this cafe WCD becomes a bended lever, 
 and the power P adls perpendicular to CD at D ; 
 and (Ax. 12.) it is all one whether the power afts 
 at D or A, in the line of direftion AD. And hence. 
 
 Cor. 2. If any force he applied to a lever ACW 
 e,t A, its power to turn it about the center of motion 
 C, is as the fine of the angle of application CAD. 
 
 For if CA be given, CD is as the fine of CAD. 
 
 Cor. Q^. In a freight lever, of thefe three, the 
 power, the weight, and the preffure upon the fupport \ 
 the middlemoft is equal to the fum of the other two. 
 
 For the middle one adts againft both the others 
 and fupports them. 
 
 Cor.
 
 64 MECHANIC POWERS. 
 
 Fig. Cor. 4. From the foregoing properties of the le- 
 30. ver^ the effe^s of fever al forts of fimple machines 
 may he explained -, and likeijoife the manner of liftings 
 carryings drawing of heavy bodies^ and fuch like. 
 
 26. For example, if a given weight W is to be 
 raifed by a Imall power applied at A, the end of 
 the lever AW. If we divide WA in C, fo that it 
 be as CA : C W : : as the weight W : to the power 
 P ; then fixing a prop or fuppc . at 'C or rather a 
 little nearer W ; then the power ^ with a fmall 
 addition, will raife the weight W*. 
 
 27. Again, if two men be to cai'ry a weight W, upon 
 the lever CA. The weight the rrian at A carries, 
 is to the weight the man at C carries as CW, to 
 AW. And therefore the lever or beam C A ought 
 to be divided in that proportion at W, the place 
 where the weight is to lie, according to the ftrength 
 of the men that carry it. 
 
 21. Likewife if two horfes be to draw at the fwing- 
 tree AB, by the ropes AF, BG ; and the fwing- 
 tree draws a carriage &c. by the rope CD ; then 
 the force afling at A will be to the force afting at 
 B, as BC to AC. And therefore BC ought to be 
 to AC, as the ftrength of the horfe at F, to the 
 ftreno;th of the horfe at G ; the weaker horfe hav- 
 ing the longer end. But it is proper to make the 
 crofs bar AB crooked at C ; that when the ftronger 
 horfe pulls his end more forward, he may pull 
 obliquely, and at a lefs diftance from the center ; 
 whilil the weaker horfe pulls at right angles to his 
 end, and at a greater diftance. 
 
 Again, fuch inftruments as crov.'s and handfpikes 
 are levers of the firft kind, and are very ufeful and 
 handy for raifing a great weiglit to a fmall hight. 
 Alfo fcifiars, pinchers, fnuffers, are double levers 
 of the firft kind, where the joint is tJie fulcrum or 
 fupport. The oars of a boat, the rudder of a Ihip, 
 cutting knives fixed at one end, are levers of the 
 
 fecond
 
 Sea. IV. MECHANIC POWERS. 65 
 
 fecond kind. Tongs, fheers, and the bones of Fig. 
 animals, are levers of the third kind, a ladder raif- 
 ed upright, is a lever of the third kind. A ham- 
 mer drawing out a nail is a lever of the fourth kind. 32. 
 
 The Steel Tard is nothing but a lever of the firft 
 kind, whofe mechaniim or conftru6lion is this. 
 Let AB be the beam, C the point of fufpenfion % 
 P the power, movgW^ along the beam CB. The 
 beam being fufpgnded at D, move the power Pj 
 along towards C, tiU ygu find the point O, where P 
 reduces the beam to an equilibrium. Then at A 
 hang on the weight W of i pound ; and move P 
 to be in equilibrio with it at i ; then hang on W 
 ' of 2 pound, and fliift P till it be in equilibrio, at 
 2. Proceed thus with 3, 4, 5, &c. pounds at W, 
 and find the divifions 3, 4, 5, &c. Or if you willj 
 after having found the points O, i ; make the di- 
 vifions, 12, 23, 34, &c. each equal to Oi. But 
 for more exaflnefs and expedition, having found 
 the point O, where P makes the beam in equilibrio : 
 hang on any known number of pounds, as W ; 
 and move P to the point B, where it makes an 
 equilibrium. Then divide OB into as many equal 
 parts as W confifts of pounds : mark thefe divi- 
 fions I, 2, 3, 4, &c. Then any weight W being 
 fufpended at A. If P be placed to make an equi- 
 librium therewith •, then the number where P hangs 
 will ihew the pounds or weight of W. 
 
 To prove this, we muft obferve, that AC is the 
 heavier end of the beam ; therefore let Q be the 
 Momentum at that end to make an equilibrium 
 with P fufpended at O •, that is, let Q^ zz CO X P. 
 But (Cor. 2. Prop. XXXVI.) Q^+ Ca X W - 
 CF X P zi CO X P -f OF X P. Take away Q 
 = CO X P, and then CA X W iz OF X P. 
 Whence AC : P : : OF : W. But AC and P are 
 
 always the fame ; therefore W is as OF ; that is, 
 
 F ;r ■
 
 66 MECHANIC POWERS. 
 
 Fig. if OF be 1, 2, 3, &c. divifions, then W is i, 2, 3, 
 32. &c. pounds. 
 
 We may take notice that the divifions Oi, 12, 
 23, &c. are all equal; but CO may be greater or 
 iefier, or nothing. 
 
 If you would know how much the weight P is, 
 take the diftance CA, and fet it from O along the 
 divifions O, i, 2, 3, &c. and it will reach to the 
 number of pounds. But this is of no, confequence, 
 being only matter of curiofity. 
 
 PROP. XLVI. 
 
 •33. In the compound lever ^ or where fever al levers a^ 
 upon one another, as AB, BC, CD, whofe fupports 
 are F, G, I ; the power P : is to the weight W : j 
 4is BF X CG X Dl : to AF X BG X CI. 
 
 For the power P a6ling at A : force at B : : BF : 
 AF •, and force or power at B : force at C : : CG : 
 GB *, and force or power at C : weight W : : DI : 
 
 f IC. Therefore ex equo^ power P : weight W : : 
 
 : BF X CG X DI : AF X GB X IC. 
 
 • And it is the fame thing in the other kinds of 
 
 levers, taking the refpe6tive diilances, from the fe- 
 veral props or fupports. 
 
 'prop. xLVir. 
 
 ^4. In the wheel and axle -^ the weight and power will 
 he in equilihrio, when the power P is to the weight 
 W ; as the radius of the axle CA, where the weight 
 hangs ; to the radius of the wheel CB, where the 
 power a£ls. 
 
 This is a wheel fixed to a cylindrical roller, 
 turning round upon a fmall axis j and having a 
 rope going round it. 
 
 Thro*
 
 32. T 
 
 • ' ' I 'I I ((i;; I I I g 
 ^7 ^ 2343-^(7^:910 
 
 \\ IP 
 
 Tl.K^a.tf.
 
 L 
 
 PMr.;-,;.
 
 Sea. IV. MECHANIC POWERS. &; 
 
 Thro* the center of the wheel C, draw the hori- Fig« 
 zontal line BCA. Then BP and AW are perpen- 34. 
 dicular to BA ; and BCA will be a lever whofe 
 fupport is C •, and the power adls always at the dif- 
 tance BC, and the weight at the diftance CA -, 
 which remain always the fame. Therefore the 
 weight and power adt always upon the lever BCA. 
 But by the property of the lever (Prop. XLV.) 
 BC : CA : : W : P, to have an equilibrium. 
 
 Otherwife, 
 If the wheel be fet a moving the velocity of the 
 point A or of W, is to that of B or P, as CA 
 to CB ; that is (by fuppodtion), as P : W. There- 
 fore W X velocity of W =: P X velocity of P ; 
 therefore the motions of P and W, being equal, 
 they cannot, when at reil, move one another. 
 
 Cor. T. If the power a5iv,ig at the radius CB, aol 3^, 
 not at right angles to it \ draw CD perpendicular to 
 BP the direction of the power \ then the power P : is 
 to the weight W : : as the radius of the axle C A : to 
 -the perpendicular CD. 
 
 For in the lever DC A, whofe fupport is C, the 
 power P : weight W : : CA : CD. 
 
 Cor. 1. In a roller turned rounds on the axis or 3^. 
 fpindle FC, hy the handle CBG \ the power applied 
 perpendicularly to BC at B, is to ihe weight W : : 
 as the radius of the roller DA, to the length of the 
 handle CB. 
 
 For in turning round, the point B defcribes the 
 circumference ot a circle ; the fame as if it was a 
 wheel whofe radius is CB. 
 
 Scholium. 
 All this is upon fuppofit ion that the rope fuf- 
 taining the weight is of no fenfible thicknefs. But 
 tf it is"a thick rope, or if there be fevsral folds of 
 
 Fa it
 
 6$ MECHANIC POWERS. 
 
 Fig. it about the roller or barrel -, you muft meafure 
 36. to the middle of the out fide rope to get the radius 
 of the roller. For the diftance of the weight from 
 the center is increafed fo much, by the rope's go- 
 ing round. 
 
 From hence the effefts of feveral forts of ma- 
 chines, or inftruments, may be accounted for. A 
 roller and handle for a well or a mine, is the 
 fame thing as a wheel and axle, a windlefs and 
 a capftain in a fliip is the fame ; and fo is a crane 
 to draw up goods with. A gimblet and an auger 
 to bore with, may be referred to the wheel and axle. 
 
 The wheel and axle has a particular advantage 
 over the lever ; for a weight can but be raifed a 
 very little way by the lever. But by continual 
 turning round of the wheel and roller, the weight 
 may be raifed to any hight required. 
 
 PROP. XLVIII. 
 
 ^ 7, In a combination of wheels with teeth ; if the pow- 
 er P be to the weight W : : as the product of the dia- 
 meters of all the a^les or pinions^ to the product of 
 the diameters of all the wheels ; the power and weight 
 will he in equilibrio. 
 
 AC, CD are the radii of one wheel and its axle •, 
 DG, GH, the radii of another ; and HI, IK arc 
 thofe of another. Thefe a<5b upon one another at 
 D and H, then as the power or force P is propa- 
 gated thro* all the wheels and axles to W ; wc 
 mufl proceed to find the feveral forces adting upon 
 them, by Prop. XXXVII. Thus, 
 
 CA 
 
 CD : CA : : P : ^ P zz force adling at D. 
 
 CA ^ CAxGD 
 
 a^d GH : GD : : ^ P (force at D) : cd^TGH^ 
 
 zz force aiding at H. 
 
 and
 
 Seft.IV. MECHANIC POWERS. 6g 
 
 CA X GD FW. 
 
 and IK : IH : : ^p ^ q^ P (force at H) : ° 
 
 CA X GD X IH 
 
 CD X GH X IK ^ ~ ^^^^ at K iz W. And CA X 
 GD X IH X P = CD X GH X IK X W v whence 
 P: W: iCDxGHxIK :;CAxGDxIH. 
 
 Cor. I . If the weight and power be in equilibria., 
 and made to move ; the 'velocity of the weight., is ta 
 the velocity of the power -, as the product of the dia- 
 meters of till the axles or pinions., to the produ^ of 
 the diameters of all the wheels. Or infiead of the 
 diameters., take the number of teeth in thefe axles, and, 
 wheels that drive one another. And the fame is true 
 cf wheels carried about by ropes. 
 
 For the power is to the weight ; as the velocity 
 of the weight to the velocity of the power. And 
 the number of teeth in the wheels and pinions, 
 that drive one another, are as the diameters. And 
 the ropes fupply the place of teeth. 
 
 Cor. 2. In a combination of wheels with teeth, 'The- 
 mimber of revolutions of the firft wheel., is to the num- 
 ber of revolutions of the lajl wheel., in any time j 
 as the product of the diameters cf the pinio7is or 
 axles., to the product of the diameters of the wheels : 
 or as the product of the number of teeth in the pi- 
 nions, to the prodvM of the number of teeth in the 
 wheels which drive them. And the fame is true of 
 wheels going by cords. 
 
 For as often as the number of teeth in any pi- 
 nion, is contained in the number of teeth of the 
 wheel that drives it ; fo many revolutions does that; 
 pinion make for one revolution of the wheel. 
 
 Scholium. 
 
 A pinion is nothing but a fmall wheel, fixed at 
 
 the other end of the axis, oppofite to the wheel ; 
 
 F \ and
 
 yo MECHANIC POWERS. 
 
 Fig- and confills but of a few leaves or teeth -, and there- 
 ^y. fore is comnnonly lefs than the wheel. But in the 
 fenfe of this propofition, a pinion may, if we 
 pkafe, be bigger than the v;heel. As if we put 
 the power and weight into the contrary places, the 
 wheels will become the pinions, and the pinions 
 the wheels, according to the meaning of this pro- 
 pofition. 
 
 PROP. XLIX. 
 
 If a power fujlains a weight by means of a fixed 
 ftilley ; the poiver and weight arc equal : but if the 
 pilky be movable- along with the weighty then the 
 weight is double the power. 
 
 A pulley is a fmall wheel of wood or metal, 
 turning round upon an axis, fixed in a block -, on 
 the edge of the pulley is a groove for the rope to go 
 over, 
 og^ Thro' the centers of the puUics, draw the ho- 
 rizontal lines AB, CD; then will AB reprefent a 
 lever of the firft kind, and its fupport is the cen- 
 ter of the pulley, which is a fixed point, the block 
 being fixed at F. And the points A, and B, where, 
 the power and weight a<5l, being equally diftant 
 from the fupport, therefore (Prop. XLV.)thepov.^er 
 P — weight W. 
 
 Alfo CD reprefents a lever of the fccond kind, 
 whofe fupport is at C, a fixed point •, the rope CG 
 being fixed at G. And the weight W ading at 
 the middle of CD, and the power ading at D, 
 twice the diflance from C -, therefore (Prop. XLV.) 
 the power P is to the weight W : : as ~ CD to 
 CD ; or as i to 2. 
 
 Cor. Hence all fixed pulleys are levers of the firfi 
 hnd^ and fcrve only to change the diretlion of the 
 .^notion; but make no addition at all to the power. 
 
 And 
 
 39
 
 Sed. IV. MECHANIG POWERS. yi 
 
 And therefore if a rope goes over feviral fixed fid- Fig. 
 lies; the power is not increafed, but rather decreafed^ 38, 
 by the fri5iion. oa^ 
 
 Scholium. 
 The ufe of a fixed pulley is of great fervice in 
 raifing a weight to any height, which otherwife 
 muft be carried by ftrength of men, which is often 
 impradicable. Therefore if a rope is fixed to the 
 weight at W (fig. 38.) and paiTed over the pulley 
 BA ; a man taking hold at P will draw up the 
 weight, without moving from the place. And if 
 the weight be large, feveral perfons may pull to- 
 gether at F, to raife the weight up j where in ma- 
 ny cafes they cannot come to it, to raife it by 
 strength. 
 
 P R O P. L. 
 
 In a combination of pullies^ all drawn by one rope 40. 
 going over all the pullies ; if the power P is to the 
 weight W ; as I to the number of the parts of the 
 rope proceeding from the movable block and pullies^ 
 'Then the power and weight will he in equilibria. 
 
 Let the rope go from the power about the pul- 
 lies in this order, ntovrs, where the laft part s is 
 fixed to the lower block B. Now (Ax. 13. J all 
 the parts of the rope ntovrs are equally ftretchcd, 
 and therefore each of them bears an equal weight ; 
 but the part 71 bears the power P, which goes to 
 the fixed block A. All the other parts, fuftainthe 
 weight and movable block B, each with a force 
 equal to P. Therefore P is to the fum of all the 
 forces, fuilained by <?, r, j, /, v, or the Vv'eight 
 W, as I to the number of thefe ropes immediate- 
 ly communicating with the movable block B. And 
 all the ropes having an equal tenlion, none of 
 
 F 4 them.
 
 72 MECHANIC POWERS. 
 
 Fig. them can move the reft, but they muft remain in 
 
 40. equiiibrio. 
 
 And if you take away the power at P, and ap- 
 ply a force at the rope / equal to P, to pull up- 
 wards in diredion /A -, this will make no alteration, 
 for the rope / draws from the movable block with 
 the lame force as before, and therefore the weight 
 is fuflained as before ; for the upper pulley (by 
 Prop. XLIX. Cor.) which the rope nt goes over, 
 ferves only to change the diredlion. And there- 
 fore as there are the fame number of ropes ftill 
 drawing from the movable block as before ; the 
 propofition holds good alfo in this refpeft. And it 
 would be the fame thing if the rope s was fixed to 
 the weight W inftead of the block B ; but had it 
 been fixed to the block A, there muft have been a 
 pulley more below, and a rope more, which would 
 have increafed the power, according to the propo- 
 fition. 
 
 Cor. I , Hence it appears to he a difidvantage to the 
 fower to pull againft the fixed block. 
 
 For the rope n has no more purchafe, or no more 
 effe(^ than the rope / has which draws againft the 
 movable block ; and therefore when one draws by 
 the rope n^ there muft be a pulley more, which 
 will create more fri<^lion. 
 
 Cor. 2. Hence one may explain the effects of all 
 forts of machines ccmpofed of pullies \ or find out 
 fuch a confiru^lion or combination of them as to an- 
 fwo any purpofe defired. And to find its force., be- 
 gin at the power., and call it i ; then all parts of the 
 running rope that go and return about fever al pullies., 
 wufi he each numbered alike. And any rope that a^is 
 a'-ainfi feieral others muft be numbered with the fum 
 ^ thefe. And fo on to the weight. 
 
 For
 
 Sea. IV. MECHANIC POWERS. 73 
 
 For example •, fuppofe a man wanted to draw him- Fig. 
 felf up to the top of a houfe or a church. Get a 41. 
 pulley A fixed at the top, and place another B at 
 the bottom. Let a rope be fixed to the upper 
 block A, and brought down about the pulley B, 
 and then put round the upper pulley, and fo 
 brought to the ground at H. Then if a crofs 
 ftick CD be faftened to the block B by a rope; a 
 man may get aftride of the ftick, and then draw 
 hjmfelf up by the rope H. And the power to draw 
 himfelf up, will be little more than ~ of his 
 weight. For the power at H, and the two parts 
 of the rope going about the pulley B, fuftain all 
 his weight ; and each of them fuftains one third 
 of it. 
 
 If inftead of the ftick CD, he takes a chair to 
 fit in -, then when he has drawn himfelf up to any 
 hight he pleafes, he may fix the rope H to the 
 chair, and then do any fort of bufinefs, as fet up 
 a dial, point the walls, and fuch like, as is com- 
 monly done. 
 
 Again, feveral tackles are ufed aboard a fliipj 42» 
 for hoifting goods and the like. Let A, B be two 
 blocks with pullies, the upper one being fixed, 
 and let a weight W be fufpended at the fingle pul- 
 ley and rope, one end of the rope being fixed at 
 F, and the other faftened to the movable block B. 
 This pulley and rope BCF is called a Runner. Let 
 the power be at P, call it i ; then all the ropes go- 
 ing from B to A, muft be each of them i, and 
 the rope going from the block B, ading againft 
 thefe four muft be marked 4, and the other part 
 of it CF muft alfo be 4. Laftly, the weight aft- 
 ing againft the^e two, muft be 8. And then the 
 power P is to the weight W, as i to 8. 
 
 ABCD is another tackle with a runner BAD, A 43 J 
 being a fixed pulley, the two blocks B, C, are 
 both movable. The rope DAB is fixed to the 
 
 weight
 
 fj^ MECHANIC POWERS. 
 
 Fig. weight at D, and to the block B. The rope PB 
 
 43. goes and returns about the pullies BC, and at laft 
 is faftened to the block C. Let P be the power, 
 mark it i, then the other parts of the rope between 
 the blocks, mull alfo be i apiece. Then CI aft- 
 irrg againft 3, muil be 3. And AB is 4, as itadts 
 againft 4 •, likewife AD mull be 4. Therefore the 
 whole force that fuftains the weight W is 3 and 4, 
 or 7. And the power to the weight as i to 7. 
 
 44. The following is a fort of Spanijh burton^ A and 
 F are two fixed pullies ; C and B two movable 
 ones. The rope going from the power P, goes 
 round C, B, and A, and is faftened to the block 
 B. Another rope is faftened to the block B, and 
 goes over the pulley F, and is fixed to the 
 block C. Then marking the power P, i. Then 
 each part of the rope, continued over C, B, and 
 A to B again muft be each i . Then FC muft be 
 2, as it acSts againft two parts ; and likewife the 
 other part of it FB muft be 2. Then the whole 
 that lifts the weight W, is i + i -f i -{- 2 zz 5. 
 And therefore the power is to the weight as i to 5. 
 
 The friction between the pullies and blocks is 
 fometimes confiderable. To remedy which, they 
 muft be as large as they can conveniently be made, 
 and kept oiled or greafed. 
 
 PROP. LI. 
 
 45- In the fcrew^ if the po-wer applied at E, he to 
 the weighty prejfure^ i^c. at B -, as the dijiance of 
 two threads of the fcreii\ taken parallel to the axis 
 of it, is to the circumference defcribed by the power at 
 E ; then the weight and power will be in equilibria. 
 
 A fcrew is an inft.rument confifting of two parts 
 . AB, CD, fitting into one another. AB is the male 
 fcrew, called the top or fpindie •, this is a long cy- 
 lindrical body, having its furface cut into ridges- 
 
 and
 
 PlJIVyV-
 
 piJH/.^/^.
 
 Sea. IV. MECHANIC POWERS. 75 
 
 and hollows, that run round it in a fpiral manner Fig, 
 from one end to the other, at equal diftances -, 45,. 
 thefe rifings are called threads, and fo many revo- 
 lutiojis as they make, ib many threads the fcrew 
 contains. CD is the female fcrew, or the plate, 
 thro' which the other goes ; its concavity is cut in 
 the fame manner as the male, fo that the ridges of 
 the male may exactly fit the hollows of the female. 
 By reafon of the winding of the threads, as the 
 handle EF is turned one way or the other, the 
 male AB goes further in or comes further out, of 
 the female. 
 
 Let the point E of the handle, make one revolu- 
 tion,, then the male AB; will have advanced the dis- 
 tance between one thread and another, of the fcrew. 
 Therefore if G reprefent any weight, which the 
 end B adts againft,, it will be moved thro' the 
 breadth of a thread, whilft the power moves thro* 
 a circumference whofe radius is EA. Therefore 
 the velocity of G is to the velocity of E, as the 
 breadth of a thread, to the circumference defcribed 
 by E •, that is (b)i fuppofition) as the power at E, 
 to the weight at G* Therefore E X velocity of E 
 zz G X velocity of G ; and therefore their motions 
 being equal, they will be in equilibrio. 
 
 Cor. By reafon of the fri^ion^ if any weight is 
 to he removed hy a fcrew ; the power mufi be t9 
 the wei;^ ht -, at leaf as the breadth of two threads of 
 the f? ew^ to the circumference defcribed by the power ; 
 to keep the wiight in eqiiilibrio j and mufi be much 
 more to move it. 
 
 For in the fcrew there is fo much fridlion, that 
 it will fuflain the weight when the power is taken 
 away. And therefore the fridion is as great or 
 greater than the power. And therefore the whole 
 power applied muft at leafc be doubled to produce 
 any motion. 
 
 SCHO-
 
 75 MECHANIC POWERS. 
 
 Fig. 
 
 45- 
 
 Scholium. 
 
 Screws with fharp threads have far more fnSiion 
 than thofe with fquare threads, and therefore move 
 a body with more difficulty. 
 
 The fcrew, in moving a body, ads like an in- 
 clined plane. For it is juft the fame as if an in- 
 clined plane was forced under a body to raife it ; 
 the boay being prevented from flying back, and 
 the bafe of the plane being driven parallel to the 
 horizon. 
 
 The ufe of this power is very great. It is of 
 great fervice for hxing feveral things together by 
 help of fcrew nails ; it is likewife very ufeful for 
 fqueezing or preffmg things clofe together, or 
 breaking them ; alfo for raifing or moving large bo- 
 dies. The fcrew is ufed in preffes for wine, oil, or 
 for fqueezing the juice out of any fruit. The very 
 friftion of this machine has its particular ufe, for 
 when a weight is raifed to any hight •, if the power 
 be taken away, the fcrew will retain its pofition, 
 and hirder the weight from defcending again by 
 its fridtion, without any other power to fuftain it. 
 
 Jn the common fcrew, fuch as is here fuppofed ; 
 the threads are all one continued fpiral from one 
 end to the other -, but where there are two or more 
 fpirals, independent of one another, as in the worm 
 of a jack -, you muft meafure between thread and 
 thread of the fame fpiral, in computing the power. 
 
 PROP.
 
 Sea. IV. MECHANIC POWERS. 77 
 
 PROP. LIT. 
 
 £>• 
 
 In the endlefs fcrew, where the teeth of the worm 4^« 
 cr fpindle AB, drives the wheel CD, by acting againii 
 the teeth of it. If the power applied at P, is to the 
 height W, a^ing upon the edge of the wheel at C : i 
 as the diftance of two threads or teeth, between fore 
 fide and fore fide, taken along AB ; is to the circum- 
 ference defcrihed by the pcwer P. T^hen the weight 
 and power will he in equilihrio. 
 
 The endlefs or perpetual fcrew is one that turns 
 perpetually round the axis AB ; and whofe teeth 
 fit exadtly into the teeth of the wheel CD, which 
 are cut obliquely to anfwer them. So that as AB 
 turns round, its teeth take hold of the teeth of tiie 
 wheel CD, and turns it about the axis I, and raifes 
 the weight W. ... 
 
 For by one revolution of the power at P, the 
 wheel will be drawn forward one tooth ; and the 
 weight W will be raifed the fame diftance. There- 
 fore the velocity of the power, will be to that of 
 the weight •, as that circumference, to one tooth : : 
 that is (by fuppofition) as the weight W, to the 
 power P. Therefore the power P X velocity of 
 P zi W X velocity of W ; therefore their motions 
 being equal, they will be in equilibrio. 
 
 Cor. If a weight N he fufpended at E on the axle 
 EF ; then if the power P, is to the weight N : : as 
 the breadth of a tooth X EF, to the circumference 
 defcrihed ^jy P X CD. They will be in equilibrio. 
 
 Or if the poiver P ; is to the weight N i : as ra- 
 dius of the axle EI, to the radius of the handle FP 
 X by the number of teeth in CD ; they will be in equi' 
 lihrio. 
 
 For
 
 7? MECHANIC POWERS. 
 
 Fig. For power P : weight W : : i tooth : circumference. 
 
 46. andweightW : weight N-: : EF : CD. 
 
 therefore P : N : : i toothxEF : CD X circumference. 
 Or thus, whilft EF turns round once, P turns 
 round as oft as CD has teeth ; whence El : BP X 
 number of teeth ; : velocity of N : velocity of P : : 
 P:N. 
 
 Scholium. 
 
 47* As the teeth of the wheel CD, muft be cut 
 obliquely to anfwer the teeth or fcrew on AB j 
 fuppofing AB to lie in the plane of the wheel CD ; 
 and therefore the wheel will be ad:ed on oblique- 
 ly by the fcrew AB. To remedy that, the fcrew 
 AB may be placed oblique to the wheel, in fuch 
 a pofition, that when the teeth of the wheel are 
 cut ftreight or perp. to its plane, the teeth of the 
 fcrew AB, may coincide with them, and fit them. 
 By that micans the force will be directed along the 
 plane of the wheel CD. Fig. 47 explains my 
 meaning. 
 
 This machine is of excellent ufe, not only in it- 
 fclf, for raifmg great weights, and other purpofes ; 
 but in the conftrudion of feveral forts of compound 
 engines. 
 
 PROP. LIII. 
 
 48. I^ i^^ wedge ACD, // a power a^ing perpendicu- 
 lar to the back CD, is to the force a5Hng againfi 
 either fide AC, in a direction perpendicular to it •, as 
 the hack CD, to either of the fides AC ; the wedge 
 will he in equilihrio. 
 
 A wedge is a body of iron or fome hard fub- 
 ftance in form of a prifm, contained between two 
 ifoceles triangles, as CAD. AB is the hight, and 
 CD the back of it; AC, AD the Hdes. 
 
 Let AB be perp. to the back CD, and BE, BF, 
 perp. to the Iides AC, AD. Draw EG, FG pa- 
 
 rall^
 
 Sea. IV. MECHANIC POWERS. 79 
 
 rallel to BF, BE •, then all the fides of the paral- Fio-. 
 lelogram BEGF are equal. The triangles EGB, 48, 
 ADC are fimilar-, for draw EOF which will be 
 perp. to AB ; then the right angled triangles AEB, 
 AEO, are fimilar, and the angle ABE zz AEO z= 
 ACB -, that is, GBE zz ACD, and likewife BGE 
 =: ADC, whence CAD = BEG. 
 
 Now let BG be the force ading at B, in direc- 
 tion BA, perp. to CD •, then (Prop. IX.) the forces 
 againft the fides AC, AD, will be in the diredions 
 EB, FB ; and therefore EB, EG will reprefent 
 thefe forces (by Prop. VIII.), when they keep one 
 another in equihbrio. Therefore force BG appli- 
 ed to the back of the wedge, is to the force BE, 
 perp. to the fide AC ; as BG to BE ; that is, (by 
 fimilar triangles) as CD to CA. 
 
 Cor. 1 . The power a^ing againjh the back at B, is 
 to that part of the force agairfi AC, which a5is pa- 
 rallel to the hack CD \ as the hack CD, is to the 
 bight AB. ' 
 
 For divide the whole force BE into the two BQ, 
 OE ; the part EO a6ts parallel to CD ; therefore 
 the force acting at B, is to the force in direclion 
 OE or BC -, as BG to OE 5 that is, (by fimilar tri- 
 angles) as CD to AB. 
 
 Cor. 2. By reafon of the great fri^ion of the 
 isoedge^ the power at B, mufi be to the rejijlancs 
 againft one Jide AQ; at leafi as twice the bafe CD, 
 to the fide AC, taking the reftflance perp. to AC. 
 Or as twice the lafe CD, to the hight AB, for the 
 refffiance parallel to the bafe CD ; to overcome the re- 
 fiflance. But the power niufi be doubled for the rejif- 
 tance againjl both Jides. 
 
 For fince the wedge retains any pofition it is 
 driven into -, therefore the fridion rnuft be. at leaft: 
 equal to tlie power that drives it. 
 
 Cor.
 
 to MECHANIC POWERS. 
 
 Fig. Cor. 3. If you reckon the refifiance at loth fidei 
 48. of the wedge \ then, if there is an equiiilrium, the 
 fower at B, is to the whole refifiance \ as the back 
 CD, to the fum of the fides, CA, AD, reckotiin^ the 
 refifiance perp. to the fides. Or as the hack CD, to 
 twice the bight AB, for the refifiance parallel to the 
 hack CD. 
 
 This follows direftly from the Prop, and Cor. i. 
 
 Scholium. 
 
 The principal ufe of the wedge is for the cleav- 
 ing of wood or feparating the parts of hard bo- 
 dies, by the blow of a mallet. The force im- 
 prefled by a mallet is vaftly great in comparifon of 
 a dead weight. For if a wedge, which is to 
 cleave a piece of wood, be prefled down with 
 never fo great a weight, or even if the other me- 
 chanical powers be applied to force it in ; yet the 
 effefl of them will fcarce be fenfible •, and yet the 
 ftroke of a fledge or mallet will force it in. This 
 cffefb is owing very much to the quantity of miO- 
 tion the mallet is put into, which it communicates 
 in an inftant to the wedge, by the force of percuf- 
 fion. A great deal of the refiftance is owing to 
 friction, which hinders the motion of the wedge i 
 but the flroke of a mallet overcomes it -, upon 
 which account the force of percufllon is of excel- 
 lent ufe ; for a fmart ftroke puts the body into a 
 tremulous vibrating motion, by which the parts 
 are difunited and feparated ; and by this means the 
 fridion or fticking is overcome, and the motion of 
 the wedge made eafy. 
 
 1 his mechanic power is the fmipleft of any ; 
 and to this, may be reduced all edge tools, as 
 knives, axes, chilTels, fcilfars, fwords, files, faws, 
 fpades, fhovels, &c. which are fo many wedges 
 faftened to a handle. And alfo all tools or inllru- 
 
 mentf;
 
 Seft. IV. MECHANIC POWERS. 8i 
 
 ments with a fharp point, as nails, bodkins, nee- Fig. 
 dies, pins •, and all inftruments to cleave, cut, Qit, 48. 
 chop, pierce, bore, and the like. And in general 
 all inftruments that have an edge or point. 
 
 This Prop, is the fame as Prop. XXXo in my 
 large book of Mechanics, but demonitrated after 
 a different way ; and both come to the fame thing, 
 which evinces the truth thereof. 
 
 In this Prop. I have fliev/n under what circum- 
 ftance, the wedge is in equilibrio •, and that is, 
 v/hen the power is to the force againfb either fide ; 
 as the back, is to that fide. Therefore it mufl be 
 very ftrange, that any body ihould underftand ir, 
 as if I had faid, that the power is to the whole 
 refiftance •, as the back, is to one fide only. They 
 that do this muft be blind or very carelefs. 
 
 a ■ SECT.
 
 [ 82 ] 
 Fig. 
 
 'D* 
 
 SECT. V. 
 
 Tl:)e comparative Strength of Beams of 
 7'i?nber, and the Strefs they fujiain. 
 The Powers of Engines ^ their Motions ^ 
 and FriBion, 
 
 PROP. LIV. 
 
 49. Jf a beam of wood AB, whofe [eSlion is a paral- 
 lelogram^ be fupported at the ends A and B, by two 
 props C, D. And a iz'eight E be laid on the middle 
 of it, to break it ; the flrength of it ivill be as the 
 fqttare of the depth EF, ivhen the breadth is given. 
 
 For divide the depth EF into an infinite num- 
 ber of equal parts at ;/, o, />, q, r, &c. Now the 
 ftrength of the beam confills of the llrength of 
 all the fibres F«, no, op, &c. And to break thefe 
 fibres, is to break the beam. Alfo when the beam 
 is itretched by the weight, the fibres Fn, no, op, 
 &c. are ftretched by the power of the bended le- 
 vers AEF, AE;/, AE^, &:c. whofe fupport is at 
 E, and power at A. For the prefllire at A being 
 half the weight E, we muft fuppofe that preHure 
 applied to A, to overcome the refillan ces at F, «, 
 €, &c. Put the force or prelTure at A zz P, then 
 P acts againfl: all the fibres at F, n, 0, &c. by 
 help of the bended levers AEF, AE;z, AEo, &c. 
 But it is a known property of fprings, fibres, and 
 fuch like expanding bodies ; that the further they 
 are ftretched, the greater force they exert, in pro- 
 portion to the length. Therefore when the beam 
 breaks j that is, when the tenfion of the fibre F» 
 
 is
 
 Sedl.V. STRENGTH OF TIMBER. 83 
 
 ,is at its utmofl extent; then thofe in the middle Fig. 
 between F and E will have but half the tenfion, r:nd 49. 
 thole at all other diftances, will have a tenfion pro- 
 portional to that dillance. This being fettled, let the 
 utmoft tenfion of nF be ~ i ; then the tenfions at ?/, 
 
 En Eo Ep 
 0, p, &c. will be gp' j-po £P &c. and the feve- 
 
 ral forces, thefe exert againfl the point A, by means 
 of the bended levers FEA, «EA, oEA, &c. will 
 FE En' Eo' Ep' 
 
 ^' EA' kF7 EA' EF3^"a' EF3r£A^" ^"^- 
 
 I 
 
 the fum o£ all is — £p v. £^ X into EF' + E«' 
 
 + E<?' + Ep^- &c. to o. But (Arith inf. Prop. 
 III.) the fum of the progrelnon tF' + En' + Eo' 
 &c. to o, is 4- EF^ Therefore the fum of all the 
 
 1 
 forces exerted at A ; that is, P zz fm x FA ^ 
 
 EF'- 
 4- EFj -zz T^T* But P =: 4 weight E, therefore 
 
 EF.^ 
 weight E = * X p-T' when the beam breaks. 
 
 In like manner for any other depth E^, the weight 
 
 • • ^Ej)" 
 
 e that would break it is zz ""jTT* Whence the 
 
 weight E to the weight e, is as EF^ to Ep' ; that 
 is, as the fquares of the depths ; for ^ p . is a given 
 
 quantity. Therefore the ftrength of the beams, 
 are as the fquares of the depths. ' 
 
 Cor. r. Hence the fir engths of fever al pieces of the 
 farae timber^ are to one another as the breadths and 
 fq_uares of the depths. 
 
 For by this Prop, they are as the fquares of the 
 
 depths when the breadth is given. And if the 
 
 breadth be increafed in any proportion, it is evi- 
 
 G 2 dent
 
 ^«4 STRENGTH OF TIMBER. 
 
 Ic'ig. dent the ftrcngth is increafed in the fame propor- 
 
 49. tion. So that a beam of the fame depth being 
 twice as broad is twice as ftrong, and thrice as 
 broad is thrice as ftrong, &c. 
 
 50. Cor. 2. If fever al beams of timber as AF of the 
 fame leri'^th^ jlick out of a wall ; their firength to 
 bear any weight W fufpended at the end, is as the 
 breadth and fquare of the depth. 
 
 This follows from Cor. i. only turning the beam 
 upfide down, to make the weight W fufpended at 
 A a<5t downwards inftead of prefiing upwards. 
 
 49. Cor. 3. If fever al pieces of timber be laid under 
 one another^ they will be no flronger, than if they 
 w-ere laid Jide by fid^. 
 
 For not being connected together in one folid 
 piece, they can only exert each its own ftrength, 
 •which will be the fame in any pofition. 
 
 Cor 4. Hence the fame piece of ti'nber is fironger 
 when laid edgeways or with the fiat Jide up and down, 
 than zvhen laid flat ways, or with the flat fide hori- 
 zontal ; and that in proportion of the greater breadth 
 to the leffcr. 
 
 For let B be the greater breadth, or the breadth 
 of the flat fide ; b the leffcr breadth, being the 
 narrow fide. Then the ftrength edge ways is BB^, 
 and fiat ways "Ebb j and they are to one another as 
 B to^. 
 
 PROP. LV. 
 
 49. Jf ^ beam of timber AB be fupported at both ends j 
 and a given weight K laid on the middle of it ; the 
 flrefs it fluff ers by the weight, will be as its length AB. 
 
 F'or half the weight E is fupported at A, by the 
 prop C j and the prefTure at C is equal to it And 
 this prelTure is always the fame whatever length 
 
 AB
 
 Seft. V. STRESS OF TIMBER. S^ 
 
 AB is of. But it was fnewn in the laft Prop, that Fig. 
 the preiTure at A, breaks the fibres Fw, no^ op, &c. 49. 
 by means of the bended levers AEF, AEw, AEo, 
 &c. But (by Prop. XLV.) when the lengths EF, 
 E;/, E^, &c, are given, and the povy-er at A alfo 
 given i the effeft at F, n, c, &:c. is fo much great- 
 er, as the arm AE is longer j that is, the ftrefs at 
 the feflion EF, is proportional to the diftance AE, 
 or to the length of the beam AB. 
 
 Cor. I . If AF be a heam Jlicking out of a ivally 5a. 
 and a weight W hung at the end of it. The flrcfs it 
 fillers by the weight, at any point G, will be as the 
 di fiance AG. 
 
 For this has the fame effecSb, as in the cafe of 
 this Prop, only turning the beam upfide down. Or 
 thus, fuppofe AHG to be a bended lever, whofe 
 fulcrum is H -, then fmce GH is given, and the 
 weight AV ; therefore by the power of the lever, 
 the longer AH is, the more force is applied at G, 
 or any other points, in GH, to feparate the parts 
 of the wood -, and therefore the ftrefs is as AG. 
 
 Cor. 2. Therefore, injlead of a weight, if any 
 force be applied at the end A, of the lever AF; 
 the Jlrefs at any part G, will be as the force, and 
 diftance AG. 
 
 For augmenting the force, the ftrefs is increafed 
 in the fame ratio. 
 
 Cor. 3. Hence alfo if any weight lie upon the 49. 
 tniddle of a horizontal beam ; the firefs there will be 
 as the weight and length of the beam. 
 
 For if the weight be increafed, the ftrefs will be ' 
 increafed proportionally, all other circumftances 
 remaining the fame. 
 
 Cor. 4. The firefs cf beams by their own weight, 
 will be as the fqiiares of the lengths. 
 For here the weight is as the length. 
 
 G 3 PROF.
 
 B6 STRESS OF TIMBER. 
 
 ^^^' PROP. LVI. 
 
 51. Jf AB be a beam of timber whofe length is given ; 
 an.i fitpported at the ends A and B •, and if a given 
 iveigbt W be placed at any point of it G. The ftrefs 
 of the beam at G, will be as the rectangle AGB. 
 
 Let the given weight be W, then (Cor. 3. Prop. 
 XLV.) the weight W is equal to the prellure at 
 both A and B. And (Cor. 2. Prop. XXXI.) pref- 
 iure at A : prelTure at B : : BG : AG, and prefl". A : 
 pref. A 4- pref. B : : BG : BG + AG ; that is, 
 pref. A : weight W : : BG : AB, therefore preflure 
 
 RC 
 at A zz -r-5-\V, and this is the force re-ading at A. 
 
 But (Prop. LV. Cor. 2.) the ftrefs at G by this 
 force a6ling at the diftance AG, is as the force 
 
 AG X BG 
 multiplied by AG ; that is, as — XjJ — X W. But 
 
 W and AB are given, and therefore the ftrefs at G 
 is as AG X ^^' 
 
 Cor. I . '^he great ejl ftrefs of a beam is when the 
 weight lies in the middle. 
 
 For the greateft redangle of the parts, is in that 
 point. 
 
 Cor. 2 . The ftrefs at any point V by a weight at 
 
 G ; is equal to the ftrefs at G, by the fame weight at W 
 
 . For when the weight is at W, the ftrefs at G is 
 
 BP 
 
 AG X GB, and the ftrefs at P zi tt^ X the ftrefs at 
 
 G iz g^ X AG X GB zz BP X AG. Again, wjien 
 
 BP 
 BG 
 
 the weight is at P, the ftrefs at P is AP X PB; 
 
 AG AG 
 
 and the ftrefs at G =z ttj X laft ftrefs zz -zri X 
 
 AP X PB = AG X PB, the fame as before. 
 
 PROP.
 
 Sea.V. STRES S OF TIMBER. %y 
 
 Fig. 
 PROP. LVir. 
 
 If the dijiance of the walls AD and BC be given ^ 52. 
 and AB, AC he two beams cf timber of equal thick- 
 nefs ; the one horizontal^ the other inclined. And if 
 two equal weights P, Q, he fiifpendsd in the middle 
 of them \ the firefs is equal in both, and the one will 
 as foon Ireak as the other^ hy thefe equal weights. 
 
 For (Prop. XIV.; AC : AB, : : weight P : 
 
 AB 
 
 -.-p P = prefTiire againfl the plane, or the part of 
 
 the weight the beam AC fuftains. And fCor. 3. 
 
 AB 
 Prop. LV.) the ilrefs upon AC is j-^ P x AC or 
 
 AB X P i and the ftrefs on AB is Q^x AB, which 
 is equal to AB X P, becaufe the weights P, Q^are 
 equal. Therefore, the ftrefs being the fame, and 
 the beams being of equal thicknefs, one will bear 
 as much as the other, and they will both break to- 
 gether. 
 
 Cor. I. If the beams he haded with weights in 
 any other places in the fame ■perpendicular line as F, 
 G ; they will hear equal firefs.^ and one will as foon 
 break as the other. 
 
 For they are cut into parts fnnilar to one ano- 
 ther; and therefore ftrefs at F : ftrefs by P : : 
 
 AB" 
 
 AFC : ^ AC^ : : AGB : : : ftrefs by B : ftrefs 
 
 4 
 by Q or ftrefs by P. Therefore ftrefs at F =z ftrefs 
 at B. 
 
 Cor. 2. If the two h earns be loaded in -proportion 
 to their lengths \ the firefs hy thefe weights., or by 
 their own weights^ will be as their lengths \ and there- 
 fore the longer., that ft and s aftope^ will fooner break. 
 G 4 For
 
 S8 STRESS OF TIMBER. 
 
 Fig. For the flrefs upon AC was AB X P, and the 
 52. ftrcfs on AB was aB X Qj, but fince P and Q are 
 
 to one another as A.C and AB, therefore the ftrefs 
 on AC and AB will be as AB X AC and AB X 
 AB •, that is, as AC to AB. And in regard to 
 their own weights, thefc are alio proportional to 
 
 their lengths. 
 
 PROP. LVIII. 
 
 53. Lei AB, AC, he two beams of timber of equal 
 leu: th and thickncfs^ the one horizontal the other fei 
 Jloping And if CD be pcrp. to AB, and they be 
 loaded in the middle ii'ith tzvo weights P, Q, which 
 are to one another as AC to AD. T'hen the flrefs 
 ivilt be equal in both, and one will as foon break as 
 the other. 
 
 AD 
 
 For (Prop. XIV.) AC : AD : : P : ^ P n 
 
 prefiure of P in the middle of AC. And by fup- 
 
 AD 
 
 pofition, AC : AD : : P : Q ; therefore -^^ P ~ 
 
 Q, the weight in the middle of AB. Therefore 
 the forces in the middle of the two beams are the 
 fame •, and the lengths of the beams being the fame, 
 therefore (Prop. LV.) the ftrefs is equal upon both 
 of them •, and being of equal thickncfs, if one 
 breaks the other v/ill break. 
 
 Cor. If the weights P, Q, be equal upon the two 
 equal beams AB, hC. 1'ke flrefs upon AB will be 
 to the firefs iipai AC, as AB or AC to AD. Tht 
 fame holds m regard to their own weights. 
 
 For the weight Q^is increafcd in that proportion, . 
 
 Scholium. 
 
 Many more propcfitions relating to the flrength 
 of timber might be inferted j as for example, if a 
 
 weight
 
 Seft.V. STRESS OF TIMBER. 8^ 
 
 weight was difpofed equally thro' the length of Fig. 
 the beam AB (fig. 51. )» fupported at both ends; 5^. 
 the ftrefs in any point G, is as the reftangle AGB. 
 And the ftrefs at any point G is but half of the 
 ftrefs it would futfer, if the whole weight was fuf- 
 pended at G. Alfo if AF (fig. 50.) be a beam 
 fixed in a wall at one end, and a weight be dif- 
 perfed uniformly thro' all the length of it. The 
 ftrefs at any point G, with that weight (or with its 
 own weight, if it be all of a thicknefs), will be as 
 AG fquare, the fquare of the diftance from the 
 end. And the ftrefs at any point G by a weight 
 fufpended at A, will be double the ftrefs at the 
 fame point G, when the fame weight is difperfed 
 uniformly thro' the part AG. They that would 
 fee thefe and fuch like things demonftrated, may 
 confult my large book of Mechanics, to which I 
 refer the reader. 
 
 PROP. LIX. 
 
 If feveral piecea of t'vnhe-'' he applied to any me- 
 chanical life where Jlreugth is requ:red\ not cnty 
 the parts of the fame piece^ hut the feveral pieces in 
 regard to one another^ ought to br fo adjujled for big- 
 nefs ; that the firength may he always proportional to 
 the firefs they are to endure. 
 
 This Prop, is the foundation of all good Mecha- 
 nifm., and ought to be regarded in all forts of 
 tools and inftruments we work with, as well as in 
 the feveral parts of any engine. For who that is 
 wife, will overload himfelf with his work tools, or 
 make them bio-o-er and heavier than the work re- 
 iquires r neither ought they to befo flender as not to 
 be able to perform their office. In all engines, it muft 
 be confidered what weight cverv beam is to carry, 
 and proportion the ftrength accordingly. All le- 
 vers
 
 go STRESS OF TIMBER. 
 
 Fig. vers muft be made ftrongeft at the place wlierc 
 they are ftrained the moft ; in levers of the lirft 
 kind, they iTiuft be ftrongeft at the fupport. In 
 thofe of the fecond kind, at the weight. In thofe 
 of the third kind, at the power, and diminifti pro- 
 portionally from that point. The axles of wheels 
 and pullies, tiie teeth of wheels, which bear great- 
 er weights, or a6t with greater force, muft be made 
 ftronger. And thofe lighter, that have light work 
 to do. Ropes muft be fo much ftronger or weaker, 
 as they have more or lefs tenfion. And in general, 
 all the parts of a machine muft have fuch a degree of 
 ftrength as to be able to perform its office, and no 
 more. For an excefs of ftrength in any part does 
 no good, but adds unneceifary weight to the ma- 
 chine, which clogs and retards its motion, and 
 makes it languid and dead. And on the other hand, 
 a defeft of itrength where it is wanted, will be a 
 means to make the engine fail in that part, and go 
 to ruin. So neceflary it is to adjuft the ftrength ta 
 the ftrefs, that a good mechanic will never negled: 
 it ; but Vy'ill contrive all the parts in due proportion, 
 by which means they will laft all alike, and the 
 whole machine will be difpofed to fail all at once. 
 And this will ever diftinguifti a good mechanic 
 from a bad one, who either makes fome parts fo 
 deft ctive, imperfeft and feeble as to fail very foon ;, 
 or makes others, fo ftrong or clumfey, as to out 
 laft all the reft. 
 
 From- this general rule follows 
 
 Cor. I. In fever al pieces cf timber of the fame 
 fort^ or in different parts of the fame piece •, the breadth 
 multiplied by the fquare of the depths muji be as the 
 length multiplied by the weight to be born. 
 
 For then the ftrength will be as the ftrefs. 
 
 Cor. 2.. The breadth mutiplied by the fquare of the 
 depths and divided by the produ^ of the length and 
 nzeight^ muft be the fame in all. Cor.
 
 Sea.V. STRESS OF TIMBER. 91 
 
 Cor. 3. Hence may be computed the ftrength of tim- Fig. 
 her proper for feveral ufes in building. As, 
 
 1. To find the dimenfions of joiils and boards 
 
 for flooring. Let b, d, I be the breadth, depth 
 
 and length of a joifl, n — number of them, x — 
 
 their diftance, g zz depth of a board, w zz weight ; 
 
 then nbdd ~ ftrength of all the joifts, and tvl zz 
 
 ftrefs on them, aifo nlgg zz ftrength of the boards, 
 
 . n r 1 r nbdd nlgg 
 and wx their ftrefs ; therefore - -j zz - ; and x 
 
 wl wx ^ 
 
 11 (ra- 
 zz -TT-p for the diftance of the joifts, or the length 
 
 of a board between them. Or ^ zz -j-r-> or dd zz 
 
 aax 
 
 llgg 
 
 -T— 5 and fo on, according to what is wanted. 
 
 2. To find the dimenfions of fquare timber for 
 the roof of a houfe. Let r, j, / be the length of 
 the ribs, fpars and lats, fo far as they bear ; a*, ^, 2 
 their breadth or depth, n the diftances of the lats, 
 w zz weight upon a rib, c zz cofine of elevation 
 of the roof. Then by reafon of the inclined plane, 
 
 Iw r . , ^^i"^ 
 
 — Xczi weight upon a fpar. And — zz weight 
 
 upon a lat : for the ribs and lats lie horizontally^ 
 
 X^ y5 2' 
 
 Therefore (Cor. 2.) — zz -j-^ zz ; — . 
 
 r rs 
 
 rrv^ rrsz^ 
 
 Whence x'^ zz —]--> and x^ zz ~7;~* Hence if any 
 
 one X, J, or z be given, and all the reft of the 
 quantities ; the other two may be found. Or in 
 general, any two being unknown, they may be 
 found, from having; the reft given. 
 
 For example, let r zz 9 feet, j zz 4 feet, / zz 
 15 inches, n zz 11 inches, c zz .707 the cofine of 
 45% the pitch of thereof. And afllime jy zr 24. 
 
 inches -,
 
 92 STRESS OF TIMBER. 
 
 Fig. , 8 1 
 
 inches i then ;r — 2 4- x/— — — rz 7.rinches. And 
 
 6-565 
 
 54. 3. To find the curve ACB, into the form of 
 •which, if a joill be cur, on the upper or under 
 fide ; and having the two fides parallel planes, 
 which are perp. to the horizon. That the faid joifl 
 Ihall be equally flrong every where to bear a given 
 weight, fufpended on it. 
 
 Let the v/eight be placed in the ordinate CD ; 
 and the breadth of the beam, and the weight being 
 given-, then (Prop LIV.j the llrength at C is as 
 CD\ And (Prop. LVI.) the (Irefs is as ADB. 
 Therefore that the ftrength may be as the ftrefs, 
 CD'' is as the redlangle ADB •, and therefore the 
 curve ACB is an ellipfis. 
 
 ^c^. 4. To find the figure of a beam AB, fixed with 
 one end in a wall, and having a given weight W 
 fufpended at the other end B ; and being every 
 where of the fame depth \ it may be equally ftrong 
 throughout. 
 
 Let^CD be the breadth at C ; then (Prop. LIV.) 
 the ftrength is as CD. And (Cor. i. Prop. LV.) 
 the ftrefs is as CB. Therefore CD is every where 
 as CB, and therefore CDB is a plane triangle. And 
 the beam is a prifm, whofe upper and under fides 
 are psiallel to the horizon. 
 
 A 6. 5* ^^ ^"*^ ^^^ figure of a beam A3, fticking 
 with one end in a wall, and of a given breadth ; 
 having a weight W fufpended at the end B j fo 
 that it may be equally ftrong throughout. 
 
 Let CD be the depJi at C. Then fince the 
 breadth is given, the ftrength is as CD\ And the 
 ftrefs as DB ; therefore CD' is as D^. Whence 
 CD is a common parabola. 
 
 ^ 6. To find the figure of a beam AB, of the 
 
 ^^'' fame breadth and depth, fticking in a wall with 
 » one
 
 Sea.V. STRESS OF TIMBER. 93 
 
 one end, and bearing a weight Hifpended at the Fig. 
 otherendB j fo that it maybe equally ftrong through- 57. 
 out. 
 
 Let CD be the thicknefs at O. Then the llrength 
 is as CD', and the ftrefs is as BO. Therefore BO 
 is as CD' or as CO'. And confequently ACB is 
 a cubic parabola, whofe vertex is at B. 
 
 7. In like manner, if CBD be a beam fixed with 58. 
 one end in a wall, and all the fides of it be cut 
 into the form of a concave parabola, whole vertex 
 
 is at B. It v/ill be equally ftrong throughout for 
 
 fupporting its own weight. 
 
 For putting BO =z x, CO n y, then by nature 
 
 of the curve, ay — xx. But the folidity of CBD is 
 
 Q . 1 4 1 6yyx 
 
 ■ — -• And the center of gravity I, is diftant 
 
 5 
 A X from B, therefore OI zr -V x. Now CD^ or 
 
 7.14.-1 6yyx 
 8j' = ftrength at O. And CBD X 01 or X 
 
 5 
 -i .V n: ftrefs. Therefore the ftrensth : to the ftrefs : : 
 
 2- 1 4.1 6y'' XX ^.i4.i6xx 
 is as 8j' : to ~ : : Sy : j- — : : sy - 
 
 : : 340 : 3.i4i6<2, that is, in a given ra- 
 tio. And as this happens every where, the folid 
 is equally ftrong in all parts. 
 
 I muft take notice here that the 1 1 6th figure in 
 my large book of Mechanics, is drawn wrong, it 
 ftiould be concave inftead of being convex, 
 
 8. Again, if AB be the fpire of a church which $9- 
 is a folid cone or pyramid ; it will be equally ftrong 
 throughout for refifting the wind. For the quan- 
 tity of wind falling on any part of it ACD, will 
 
 be as the feftion ACD. Therefore let AO iz r, 
 CD zz y. And x zn ay, then the ftrength at O zz 
 V', and if I be the center of gravity of ACD, then 
 
 01
 
 54 STRESS OF TIMBER. 
 
 Fig. OI z: I X. And the ftrefs at O i= wind ACD X 
 
 59. 01 — xy X T ^- Therefore the ftrength is to the 
 
 ftrefs : : as j^ : 4- ^^y • • JV7 • t ^'^ or 4 ^^tv^ : : 3 : ^^ ; 
 
 that is, in a given ratio. Therefore the fpire is 
 
 equally ftrong every where. 
 
 Scholium. 
 
 It is all along fuppofed that the timber, &c. is 
 of equal goodnefs, where thefe proportions for 
 ftrength are made. But if it is otherwilc, a pro- 
 per allowance muft be made for the defect. 
 
 In thefe Propofitions, I have called every thing 
 Strength, that contributes in a direft proportion to 
 refift any force afting againft a beam to bi-eal; it ; 
 and I call Strefs, whatever weakens it in a direcSt 
 proportion. But the whole may be referred to the 
 article of ftrength -, for what I have called ftrefs 
 may be reckoned ftrength in an inverfe ratio. Thus 
 the ftrength of a piece of timber may be faid to 
 be directly as the breadth and fquare of the depth, 
 and inverfely as its length, and the weight or force ' 
 applied ; and that is equivalent to taking in the 
 ftrefs. But I had rather keep them diftmft, and 
 refer to each of them their proper eftcifls, as I have 
 along done in the foregoing examples. 
 
 A piece of wood a foot long, and an inch fquare, 
 will bear as follows-, oak from 320 to iioo; ehn 
 from 310 to 930 i fir from 280 to 770 pounds, 
 according to the goodnefs. 
 
 PRO P.
 
 Sect.V. COMPOUND ENGINES. 95 
 
 Fig- 
 PROP. LX. 
 
 In any machine contrived to raife great weights ; // 
 the power applied^ be to the weight to be raifed •, a-s 
 the velocity of the weighty to the velocity of the 
 power ; the power will only be in equilibria with the 
 weight. I'herefcre to raife it, the power mufv be fq 
 far increafed, as to overcome all the friolion and re* 
 fifiance arijing from ths engine or olherwife ; and 
 then the power will be able to raife the weight. 
 
 A man would be much miftaken, who lliall 
 make an engine to raife a great weight, and give 
 his power no greater velocity, in regard to the ve- 
 locity of the weight J than the quantity of the 
 weight has in regard to the quantity of the power. 
 For when he has done that, his weight and power 
 will but have equal quantities of motion, and 
 therefore they cannot fet one another a moving, 
 but miufl always remain at reft. It is necelTary 
 then, that he do one of thefe two things, i. That 
 he apply a power greater than in that proportion, fo 
 much as to overcome all the friftion and other ac- 
 cidental refiftance that may happen : and in fome 
 engines thefe are very great. Or 2. He muft fo 
 continue his engine, that the velocity of the power, 
 which fuppofe he has given, may be fo much 
 greater than the velocity of the weight ; as the 
 quantity of the weight, fricStion, and refiftance and 
 all together, is greater than the power. This be- 
 ing done, the greater power will always overcome 
 the lefler, and his engine will work. 
 
 If a man does not attend to this rule, he will be 
 guilty of many abfurd miftakes, either in attempt- 
 ing things that are impoITible, or in not applying 
 means proper for the purpofe. Hence it is thai:' 
 ensrines contrived for mines and water-works fo ot- 
 
 ten
 
 96 COMPOUND ENGINES. 
 
 Fig. ten fail j as they muft when either the quantity or 
 velocity of the power is too little ; or which is the 
 fame thing, when the velocity of the weight is too 
 great, and therefore would require more power 
 than what is propofed. As the weight is to move 
 flow, the confequence is, that it will be fo much a 
 longer time in moving thro' any fpace. But there 
 is no help for that. For as much as the weight to 
 be raifed is the greater, the time of raifing it will 
 be fo much greater too. 
 
 Cor. I . Hence in raifing any weighty what is gain- 
 ed in -power is loji in time. Or the time of rifing 
 thro' any hight will be fo much longer as the weight is 
 greater. 
 
 If the power be to the weight as i to 20, then 
 the fpace thro' which the weight moves will be 20 
 times lefs, and the time will be 20 times longer in 
 moving thro' any fpace, than that of the power. 
 The advantage that is gained by the ftrength of 
 the motion, is loft in the flownefs of it. So that 
 tho' they increafe the power, they prolong the time. 
 And that which one man may do in 20 days, may 
 be done by the ftrength of tv/enty men in one day. 
 
 Cor. 2 . The quantity of motion in the weight is not 
 at all increaffd by the engine. And if any given 
 quantity of power be immediately applied to a body at 
 liberty^ it will produce as much motion in it, as it 
 would do by help of a machine. 
 
 PROP. LXI. 
 
 If an engine be compofed of fever al of the Jlmple 
 mechanic powers combined together ; it will produce 
 the fame effetl^ fitting aftde fritfion % as any one Jim- 
 pie mechanic power would do, which has the fame 
 power or force of aUing. 
 
 For let any compound engine be divided into all 
 the limple powers that compofe it. Then the force 
 
 or
 
 JSea. V. COMPOUND ENGINES. 97 
 
 or power applied to the firll part, will caiife it to Fig. 
 aft upon the fecond with a new power, which 
 would be deemed the -weight, if the machine had 
 no more parts. This new power a6ling on the fe- 
 cond part, will caufe it to a6l upon the third part -, 
 and that upon a fourth, and fo on till you come 
 at the weight, which will be adled on, by all thefe 
 mediums, juft the fame as by a fimple machine 
 whofe power is equal to them ail. 
 
 Cor. I. Hence a compound machine may he made^ 
 which Jhall have the fame ^ower, as any fmgle one 
 ^ropofed. 
 
 For if a lever is propofed whofe power is 1 00 to 
 I ; two levers acting on one another will be equi- 
 valent to it, where the power of the firft is as 10 
 to I, and that of the fecond alio as 10 to i, or the 
 firfb 20 to I, and the fecond 5 to i j or any two 
 numbers, whofe produft is ioo. 
 
 Again, a wheel and axle v/hofe power is as 48 
 to I, may be refolved into two or more wheels 
 with teeth, to have the fame power •, tor exam- 
 ple, make two wheels, fo that the firfl wiieel and 
 pinion be as 8 to i, and the fecond as 6 to i. 
 They will have the fame effe6l as the fingle one. Or 
 break it into three v/heels, whofe feveral powers may 
 be 4 to I, and 4 to i, and 3 to i. 
 
 If a fimple combination of pullies be as 36 to 
 I •, you may take three combinations to aft upon 
 one another, whofe povv?ers are 3 to i, 3 to 1, and 
 4 to I. 
 
 And after the fame manner it is to be done m 
 machines more comDOunded. 
 
 And this is generally done to fave room. For 
 when an engine is to have great power, it is hard- 
 ly made of one wheel, it v/ould be fo laro;e ; but 
 by breaking it into feveral wheels, after rliis. man- 
 ner j it will go into a little room, and have the 
 H fame
 
 f)« COMPOUND ENGINES. 
 
 Fi"-. fame power as the other. All the inconvenience 
 is, it will have more friction •, for the more parts 
 afting upon one another, the more fridtion is made. 
 
 Cor. 2. Hence aljo it follows^ that in any corn-pound 
 machine^ its power is to the weight, in the compomtd 
 ratio of the fovoer to the weight in all thejimple ma-^ 
 chines that compofe it. 
 
 Cor. 3. Hence it will be no difficult matter to con- 
 trive an engiiie that Jhall c"jercoine any force or re/if- 
 tame affigned. 
 
 For if you have the quantity of power given, as 
 well as of the weight or refiftance ; it is but taking 
 any fmiple machine as a lever, wheel, &c. fo that 
 the power may be to the weight in the ratio af- 
 figned, adding as much to the weight as you judge 
 the fri6lion will amount to. When this ftmple 
 machine is obtained ; break it or refolve it into as 
 many other fimple ones as you think proper j fo 
 that they may have the fame power. 
 
 And as to the feveral fimpie macliines, it mat- 
 ters not what fort they are of, as to the power ; 
 whether they be levers, wheels, pullies, or fcrews ; 
 but fome are more commodious than others for 
 particular purpofes •, which a mechanic will find 
 out bell by practice. In general, a lever is the 
 moft ready and fimple machine to raife a weight a 
 fmall difhance ; and for further dillances, the wheel 
 and axle, or a combination of puilies ^ or the per- 
 petual fcrew. Aifo thefe may be combined with 
 one another ; as a lever with a wheel or a fcrew, 
 the wheel and axle with pullies, pullies with pul- 
 lies, and wheels with wheels, the perpetual fcrew 
 and the wheel. But in general a machine Ihould 
 coniift of as few parts as is confiftent with thepur- 
 pofe it is dcfigned for, upon account of leilcning the 
 fridion i and to make it Hill lefs, the joints mull 
 
 be
 
 Sea. V. COMPOUND ENGINES. ,99 
 
 be oiled or greafed. All parts that adt on one Fig. 
 another muft be poliihed fmooth. The axles or 
 fpindles of wheels muft not fhake in the holes, but 
 run true and even. Likewife the larger a pi^chine 
 is, if it be well executed, the better and truer it 
 will work. And large wheels and pullies, and 
 fmall axles or fpindles have the leaft fridlion. 
 
 The power applied to work the engine may be 
 men or horfes ; or it may be weight or a fpring ; 
 or wind, water, or fire ; of which one maift take 
 that which is moll convenient and cofts the leaft. 
 Wind and water are beft applied to work large en- 
 gines, and fuch as muft be continually kept going. 
 A man may a6l for a while againft a refiftance of 
 50 pounds ; and for a whole day againft ^o pounds. 
 A horfe is about as ftrong as five men. 
 
 If two men work at a roller, the handles ought 
 to be at right angles to one another. 
 
 When a machine is to go regular and uniform, a 
 heavy wheel or fly muft be applied to it. 
 
 Scholium. 
 Two things are required to make a good engi- 
 neer. I . A good invention for the fimpie and eafy 
 contrivance of a machine, and this is to be attain- 
 ed by pra<5tice and experience. 2. So much theory 
 as to be able to compute the eifed any engine will 
 have ; and this is to be learned from the principles 
 of Mechanics. 
 
 PROP. LXII. 
 
 The fri5lion or refiftance arifing by a body movino; 
 upon any furface^ is as the rcughnefs of the furface^ and 
 nearly as the weight of the body -, hut is not much in- 
 creafed by the quantity of the furface of the moving 
 body^ and is fomething greater with a greater 'velocity. 
 
 It is matter of experience that bodies meet with 
 
 a great deal of refiftance by fiiding upsn one ano- 
 
 H 2 thsr.
 
 ioo FRICTION. 
 
 Fig. ther, which cannot be entirely taken away, tho* the 
 bodies be made never fo Imooth : yet by imoothing; 
 or polifhing theif furfaces, and taking off the rough- 
 nefs of them, this refiftance may be reduced to a 
 fmall matter. But many bodies, by their natural 
 texture, are not capable of bearing a polifh ; and 
 thefe will always have a confiderable degree of re- 
 fiftance or friftion. And thofe that can be polifli- 
 ed, will have fome of this refiftance arifing from 
 the cohefion of their furfaces. But in general, 
 the fmoother or finer their furfaces,- the lefs the 
 fridion will be. 
 
 As the furfaces of all bodies are in fome degrea 
 rough and uneven, and fubjed to many inequali* 
 ties ; when one body is laid upon another, the pro- 
 minent parts of one fall into the hollows of th« 
 other i fo that the body cannot be moved forv/ard, 
 till the prominent parts of one be raifed above 
 the prominent parts of the other, which requires 
 the more force to effedb, as thefe parts are higher ^ 
 that is, as the body is rougher. And this is fimi- 
 Jar to drawing a body up an inclined plane, for 
 .thefe protuberances are nothing elle but fo many 
 inclined planes, over which the body is to be drawn. 
 And therefore the heavier the body, the more force 
 h required to draw it over thefe eminencies ; whence? 
 the fridion will be nearly as the weight of the body. 
 
 But whilft the roughnefs remains the fame, or the 
 prominent parts remain of the fame hight, there 
 V/ill always be required the fame force, to draw the 
 iiime weight. And the increafing of the furface, re- 
 taining the fame weight, can add nothing to the re- 
 fiftance on that account •, but it will make fome ad- 
 dition upon other accounts. For when one furface 
 is dragged along another, fome part of the re- 
 liftance arifes from fome parts of the moving fur- 
 face, taking hold of the parts of the other, and 
 tearing them, off.; and. this is called zvearin^. And 
 
 there*
 
 Sea. V. FRICTION. loi: 
 
 therefore this part of the fridtion is greater in a great- Figj^ 
 er furface, in proportion to that furface. There is 
 likewife in a greater furface, a greater force of co- 
 hefion, which flill adds fomething to the fridlion. 
 But the two parts of the friction, arifing from the 
 wearing and tenacity, are not increafed by the ve- 
 locity : but the other part, of drawing them over 
 inclined planes, will increafe with the velocity. So 
 that in the whole, the fridlion is fomething increaf- 
 ^ ed by the quantity of the furface, and by the ve- 
 locity, but not much. But more in fome bodies 
 than others, according to their particular texture. 
 
 Cor. I. Hence there can he no certain rule, to ejii- 
 vrate the fri^lion of bodies ; this is a matter that can 
 only be decided by expe?'iments. But it may be obferv- 
 ed, that, ceteris paribus, hard bodies will have lefs re- 
 Jiftance than fofter -, and bodies oiled or greafed, wjU', 
 have far lefs. 
 
 For the particles of hard bodies, cannot fo well 
 take hold of one another to tear themfelves off. 
 And when a furface is oiled, it is the fame thing^ 
 as if it run upon a great number of rollers or fpheres. 
 
 Cor. 2. Hence alfo a method appears of meafurin^ 
 the friction of a body fliding upon another body, b^: 
 help of an inclined plane. 
 
 Take a plank CB of the fame matter, raife it at ^cu 
 one end C fo high, till the body whofe friction is 
 fought, being laid at C, fhall juft begin to move 
 down the plane CB. Then the weight of the bo- 
 dy, is to the fridion as the bafe AB, to the highc 
 AC of the plane. For the prelTure againft the 
 plane is the part of the weight that caufes the 
 fridtion, and the tendency down the plane is equal 
 to the friftion. And (Prop. XIV.) that prelTure is 
 CO the tendency as AB to AC. 
 
 H3 tf
 
 lo^^ FRICTION. 
 
 Fig. If you piiOi the body from C downward, and 
 60. obferve it to keep the fame velocity thro' D to B -, 
 then you will have the friflion for that velocity. If 
 it increafes its velocity, lower the end of the plank- 
 C •, if it grows (lower, raife the end C, till you 
 get the body to have the fame velocity quite thro* 
 the plane. And fo you will find what elevations 
 are proper for each velocity ; and from thence the 
 ratio of AB to AC, or of the weight to the friftion. 
 There is a way to make the experiment, by draw- 
 ing the body along a horizontal plane, by weights 
 hung at a firing, which goes over a pulley ; but the 
 method here defcribed is more eafy and fimple. 
 
 Scholium. 
 
 From what has been before laid down, it will be 
 cafy to underftand the nature of engines, and how 
 to contrive one for any purpofe affigned. And 
 Ijkewife having any engine before us, we can by 
 t^e fame rules, compute its pov/ers and operations. 
 
 Engines are of various kinds ; fome are fixed in 
 a particular place, vviiere they are to a6l ; as wind- 
 mills and water-mills for corn, fire engines for 
 drawing water, gins for coal pits, many forts of 
 mills •, pumps, cranes, &:c. others are movable 
 frorn one place to another, and may be carried to 
 any place where they are wanted, as blocks, pul- 
 iies and tackles for raifing weights, the lifting 
 jack, and lifting {lock, clocks, watches, fmall bel- 
 lows, fcales, fleelyards, .and an infinite number of 
 others. Another fort of engines are fuch as are 
 made on purpofe to move from one place to ano- 
 ' ther, fuch as boats, fliips, coaches, carriages, wag- 
 gons, &c. If any of thefe are urged forward by 
 the help of levers, Y»'heels, &c. By having the 
 adling power given, the moving force that drives 
 it forward, is eafily found by the properties of 
 tRefe machines. Only obferve, if the firft adling 
 
 power
 
 VlK/^ajos.
 
 o^ 
 
 ris-.4j 
 
 n
 
 ^ea.V. WHEEL CARRIAGES. 103 
 
 power be externa], as wind, water, horfes, &c. Fig, 
 you muft not forget to add or fubtrad it, to or 
 from the moving force before found ; according as 
 that firft acting power confpires- with, or oppofes 
 the- motion of the machine ; and the refult is the 
 true force it is driven forward with- I have only- 
 room to defcribe a very few engines, but thofe that 
 defire. it may fee great variety in my large book of 
 Mechanics. 
 
 A WHEEL CARRIAGE. 
 
 AB is a cart or carriage, going upon two wheels 61. 
 as CD, and fometimes upon four, as all waggons 
 do. The advantages of wheel carnages is fo great, 
 that no body who has any great weight to carry, 
 will make ufe of any other method. Was a great 
 weight to be dragged along upon a fledge or any 
 ilich machine without wheels, the fridlion would be 
 fo great, that a fufficient force in many cafes could 
 not be got to do it. But by applying wheels to carri- 
 ages, the friftion is almoft all of it taken away. 
 And this is occafioned by the wheels turning round 
 upon the ground, inftead of dragging upon it. And 
 the reafon of the wheel's turning round is the refif- 
 tance the earth makes againft it at O where it 
 touches. For as the carriages goes along, thq wheel 
 meets with a refillance at the bottom O, where it 
 touches the ground ; and meeting with none at the 
 top at C, to balance it ; that force at O mull make 
 it turn round in the order ODC, fo that all the 
 parts of the circumference of the wheel are fuccef- 
 fively applied to the earth. In going down a fleep 
 bank it is often neceffary to tie one wheel faft, that 
 it cannot turn round, this will make it drag ; and 
 by the great refiftance it meets with, flops the too 
 violent motion, the carriage would otherwife have, 
 in defcending the hill 
 
 H 4 But
 
 104 WHEEL CARRIAGES. 
 
 Ficr. But altho' all forts of wheels very much dimi- 
 61. nifh the fridion -, yet fome have more than others; 
 and it may be obferved that great wheels, and fmall 
 axles have the leaft fridion. To make the friction 
 as little as poflible, fome have applied fridion 
 wheels, which is thus ; EG is the fridion wheel 
 running upon an axis I v/hich is fixed in the piece 
 of timber ES, which timber is fixed to the fide ot 
 the carriage. KL is the axle of the carriage, 
 which is fixed in the wheel CD, fo that both 
 turn round together. Then inftead of the carri- 
 age lying upon the axle KL, the fridion wheel 
 FG lies upon the axle ; fo that when the wheel CD 
 turns round, the axle caufes the fridion wheel, 
 with the weight of the carriage upon it, to turn 
 round the center I, which diminifhes the fridion in 
 proportion to the radius IG : and there is the fame 
 contrivance for the wheel on the other fide. But 
 the Vv-heel CD need not be fixed to the axle ; for 
 it may turn round on the axle KL, and alfo the 
 axle turn round under the carriage. 
 
 In paifing over any obftacles, the large wheels 
 have the advantage. For let MN be an obftacle ; 
 then drawing the wheel over this obftacle, is the 
 fame thing as drawing it up the inclined plane MP, 
 which is a tangent to the point M ; but the greater 
 the wheel CD is, the lefs is that plane inclined to 
 the horizon. 
 
 Likewife great wheels do not fink fo deep into 
 the earth as fmall ones, and confequently require 
 lefs force to pull them out again. 
 
 But there are difadvantges in great wheels ; for 
 in the firft place, they arc more eafily overturned ; 
 and fecondly, they are not fo eafy to turn with, in 
 a ftrait road as fmall v.'heels. 
 
 The tackle of any carriage ought to be fo fixed, 
 that the horfe may pull partly upwards, or lift, as 
 ■well as pull forwards ; for all hills and inequalities 
 
 in
 
 Sea. V. HAND MILL. 105 
 
 in the road, being like fo many inclined planes. Fig. 
 the weight is moit eafily drawn over them, when 61. 
 the power draws at an equal elevation. 
 
 A carriage with four wheels is more advantage- 
 ous, than one with two only, but they are bad t» 
 turn ; and therefore are obliged to make ufe of 
 fmall fore-wheels. Broad wheels which are lately 
 come into falhion, are very advantageous, as they 
 fink but little into the earth. But there is a dif- 
 advantage attends them, for they take up fuch a 
 quantity of dirt by their great breadth, as fenfibly 
 retards the carriage by its v/eight, and the like may 
 be faid of their own weight. 
 
 The under fide of the axle where the wheels are, 
 muft be in a right line ; otherwife if they flant up- 
 wards, the weight of the carriage will caufe them 
 to work toward the end, and prefs againft the 
 runners and lin pin. And as the ends of the axle 
 are conical, this caufes the wheels to come nearer 
 together at bottom, and be further diftant at the 
 top ; by which means the carriage is fooner over- 
 turned. To help this, the ends of the axle muft 
 be made as near a cylindrical form as poflible, to 
 get the wheels to fit, and to move free. 
 
 A HAND MILL. 
 
 Fig. 62. is a hand mill for grinding corn. A, B 62. 
 the flones included in a wooden cafe. A the up- 
 per flone, being the living or moving ftone. B the 
 lower flone, or the dead flone, being fixed immov- 
 able. The upper flone is 5 inches thick, and a 
 foot and three quarters broad ; the lower flone is 
 broader. C is a cog-wheel, with 16 or 18 cogs ; 
 DE its axis. F is a trundle with 9 rounds, fixed 
 to the axis G, which axis is fixed to the upper 
 ftone A, by a piece of iron made on purpofe. H 
 is the hopper, into which the corn is put ; I the fhoe, 
 to carry the corn by little and little thro' a hole at
 
 io6 HAND MILL. 
 
 Fig. K, to fall between the two Hones. L is the mill 
 62. eye, being the place where the flour or meal comes 
 out after it is ground. The under ftonc is fup- 
 ported by ftrong beams not drawn here. And the 
 fpindle G Hands on the beam MN, which lies up- 
 on the bearer O, and O lies upon a fixed beam at 
 . one end, and at the other end has a firing fixed, 
 and tied to the pin P. The underdone is not flat, 
 but rifes a little in the middle, and the upper one 
 is a little hollow. The fl:ones very near touch at 
 the out fide, but are wider towards the middle to 
 let the corn go in. 
 
 When corn is to be ground, it is put into the 
 hopper H, a little at a time, and a man turns the 
 handle D, which carries round the cog-wheel C, 
 and this carries about the trundle F, and axis G, 
 and Hone A. The axis G is angular at K ; and 
 as it goes round, it fliakes the fiioe I,, and makes 
 the corn fall gradually thro' the hole K. And the 
 upper ftone going round grinds it, and when 
 ground it comes out at the mill eye L, where there 
 is a fack or tub placed to receive it. Another han- 
 dle may be made at E like that at D, for two men 
 to work, if any one pleafes. In order to make the 
 mill grind courfcr or finer, the upper ftone A may 
 be lowered or raifed, by means of the Airing going 
 from the bearer O •, for turning round the pin P, 
 the firing is lengthened or fiiortened, and thereby 
 the timbers O, M are lowered or raifed, and with 
 them the axle G and fl:one A. For the fpindle G 
 goes thro' the ftone B, and runs upon the beam 
 MN. The fpindle is made fo clofe and tight, by 
 wood or leather, where it goes thro' the under ftone, 
 that no meal can fall thro'. The under fide of the 
 upper ftone is cut into gutters in the manner repre- 
 fentcd at Q. It is a pity fome fuch like mills are 
 not made at a cheap rate for the fake of the poor, 
 who are much diftreflfed by the roguery of the 
 millers. Fig,
 
 I 'Sf^pd.ioS.
 
 r.
 
 Sea. V. THE CRANE. 107 
 
 Fig. 6^. is a fort of crane, BC an upright poft, Fig. 
 AB a beam fixed horizontally at top of it ; thefe 63. 
 turn round together on the pivot C, and within 
 the circle S, which is fixed to the top of the frame 
 PQ. EF is a wooden roller, or rather a roller 
 made of thin boards, for lightnefs, and all nailed 
 to feveral circular pieces on the infide. GH a 
 wheel fixed to the roller, about which goes the rope 
 GR. IK, LN, two other ropes ; fixed with one 
 end to the crofs piece AB, and the other end to 
 the roller EF. W a weight equal to the weight 
 of the wheel and roller, which is faflened to a rope 
 which goes over the pulley O, and then is faflen- 
 ed to a collar V, which goes round the roller. ET 
 is another rope with a hook at it to lift up any 
 weight, the other end of the rope being fixed to 
 the roller •, here are in all five ropes. 
 
 To raife any weight as M, hang it upon the 
 hook T, then pulling at the rope R which goes 
 about the wheel GH, this caufes the wheel and 
 roller to turn round, and the ropes IK, LN to 
 wind about it, by which means the wheel and axle 
 rifes -, and by rifing, folds the rope TE about the 
 roller the contrary way, and fo raifes the weight 
 M. When the weight M is raifed high enough, a 
 man mufl take hold of the rope T with a hook^ 
 by which the whole machine may be drawn about, 
 turning upon the centers C and S. And then the 
 vv'^eight M may be let down again. The weight of 
 the wheel and roller do not affeft the pov/er draw- 
 ing at R, becaufe it is balanced by the weight W. 
 There is no friftion in this machine but what is 
 occafioned by the collar V, and the bending of the 
 ropes. And the power is to the weight in this 
 crane, as the diameter of the roller to the radius 
 of the wheel GH. 
 
 An
 
 toS FULLING M I L L, 5cc, 
 
 "^ An ENGINE for raifing Weights. 
 
 64. Fig. 64. is an engine compofed of a perpetual 
 fcrew AB, and a wheel DE with teeth, and a Tin- 
 gle pulley H. f G is an axle, about which a rope 
 goes, which lifts the pulley and weight W. BC it 
 the winch, to turn it round withal. As the fpin- 
 dle AB is turned about, the teeth of it takes the 
 teeth of the wheel DE, and turns it about, toge- 
 ther with the axle FG, which winds up the rope, 
 and raifes the pulley H, with the weight W. The 
 power at C, is to the weight W, as diameter FG 
 X by the breadth of one tooth, is to twice the di- 
 ameter DE X circumference of the circle defcrlbed 
 byC. 
 
 A FULLING MILL. 
 
 65. Fig. 65. is a fulling mill. AB a great water 
 wheel, carried about by a ftream of water, com- 
 ing from the trough C, and falling into the buck- 
 ets D, D, D whofe weight carries the wheel about; 
 this is a breaft mill, becaufe the water comes no 
 higher than the middle or breaft of the wheel ; 
 EF is its axis; I, I; K, K, two lifters going 
 thro' the axle, which raife the ends G, G of the 
 wooden mallets GH, GH, as the wheel goes about ; 
 and when the end G flips off the cog or lifter K 
 Or I, the mallet falls into the trough L, and each 
 of the mallets makes two ftrokes for one revolu- 
 tion of the wheel. The mallets move about the 
 centers M, M. Thefe troughs L, L, contain the 
 ftuff which is to be milled, by the beating of the 
 mallets. N, N, is a channel to carry the water, 
 being juft wide enough to let the wheel go round. 
 And the wheel may be ftopt, by turning the 
 trough C afide, which brings the water. In this 
 engine more mallets may be ufed, and then more 
 pins or lifters muft be put thro' the axis EF. 
 
 Fig,
 
 1 
 
 i 
 
 Tl.pa2oS.
 
 Fl TIz-^J^ 
 
 r
 
 Sea.V. A WATCH. lo^ 
 
 Fig. 66. is a common Pocket IVatch, AA the Fig. 
 halance, BB the I'^r^^i C, C, two palats. D the 66. 
 crown wheel ading againft the palats C, C ; E its 
 pinion. F the contrate wheels G its pinion. H the 
 third wheels I its pinion. K the fecond wheel or 
 center wheels L its pinion. M the great wheels N- 
 the fufee turning round upon the Ipindle of M. 
 O the fpring box^ having a fpring included in it» 
 PP the chain going round the fpring box O, and 
 the fufee N. This work is within the watch be- 
 tween the two plates. Here the face is downward, 
 and in the watch the wheel K is placed in the cen- 
 ter, and the others round about it. Here I have 
 placed them fo as befl to be feen, which fignifies 
 nothing to the motion. The balance AA is with- 
 out the plate, covered by the cock X. The mi- 
 nute hand Q goes upon the axis of the wheel K. 
 
 Then between the upper plate and the face, we 
 have V the cannon pinion or pinion of report. Z the 
 dial wheel. T the minute wheel. S the pinion or 
 nut, fixed to it. The focket of the cannon pinion 
 V goes into the focket of the wheel Z, and are 
 movable about one another, and both go thro' the 
 face ; on the focket of the pinion Z is fixed the 
 hour hand R -, and on the focket of V is fixed the 
 minute hand Q, Likewife the focket of V is hol- 
 low, and both go upon the arbour of the wheel K, 
 which reaches thro' the face, and are faflened there. 
 The wheel and focket, T, S are hollow, and go 
 upon a fixed axle on which they turn round. 
 
 When the chain PP is wound up, upon the fu- 
 fee N ; the fpring included in the box O, draws 
 the chain PP, which forces about the wheel M, the 
 fufee being kept from flipping back, by a catch 
 on purpofe. Then M drives L and K, and K 
 drives I, and H drives G, and F drives E, and 
 the teeth of the crown wheel D, acl againft the 
 palats C, C alternately, and cau'e the balance A A to 
 
 vibrate
 
 110 A W A T C H. 
 
 Fio-, vibrate back and forward, and thus. the watch is 
 kept going. 
 
 65. The cannon pinion, and dial wheel V and Z, 
 and the hands Q, R, being put upon the arbor 
 of K at W i ancl faftened there, by means of a 
 flioulder which is upon the axis, and a brafs fpring ; 
 as the wheel K goes round, it carries with it the 
 pinion V with the minute hand, and V drives T to- 
 gether with 3 •, and S drives Z with the hour hand. 
 The numbers of the wheels and pinions, (that 
 is the teeth in them) are, M — 48, L rz 12, K — 
 54, I =. 6, H = 4S, G =: 6, F zz 48, E zz 6, 
 D =z 15, and 2 palats. The irain, or number of 
 beats in an hour, is 17280, which is about 4 1. beats 
 in a fecond. Alfo V =1 10, Z zz 36, S — 12, 
 T zz 40. 
 
 The wheel M goes round 6 times in 24 hours, 
 /48n 
 therefore K goes round ( — J 4 times as much; 
 
 that is, 24 times, or once in an hour, and the hand 
 O along with it ; therefore Q will fhew minutes. 
 Then as V goes round once in an hour, T will go 
 
 round ( — J ^ of that, or ^ the circumference-, 
 
 and as S goes ^, Z will go [-^ j i- of that, or ^-V 
 
 of the circumference in an hour, and therefore as 
 R goes along with it, R will flicvv the hours. The 
 wheels and pinions T, Z, and S, V, are drawn 
 with the face upwards. And the whole machine 
 included in a cafe is but about two inches diameter. 
 There is a fpiral fpring fixed under the balance 
 AB, called the regulator, which gives it a regu- 
 lar motion ; and likewife abundance of fmall parts 
 helpful to her motion, too long to be defcribed 
 here. 
 
 The
 
 %a. V. A WATCH. Ill 
 
 The way of writing down the numbers, is thus, Fig. 
 
 48 66- 
 
 12 — 54 loQ^ 
 
 6 — 48 40 — 12 
 
 6 — 48. 36 R. 
 
 6—15 
 2 
 
 Explanation. The wheel with 48 drives a pinion 
 of 12, and a wheel of 54 on the fame arbor. The 
 wheel 54 drives the pinion 6 with the wheel 48 oa 
 the fame arbor. The wheel 48 drives the pinion 
 6 and wheel 48 on the fame arbor. The wheel 
 48 drives the pinion 6 and wheel 15 on the fame 
 arbor. And the wheel 1 5 drives the tv/o palats. 
 
 Again the wheel 54 has the pinion 10 on its ar- 
 bor, and the hand Q^; and the pinion 10 drives 
 the wheel 40, with the pinion 12. And the pini- 
 on 12 drives the wheel 0,6 with the hand R. 
 
 As this machine is moved by a fpring, it is fub- 
 je6): to very great inequalities of motion, occafion- 
 ed by heat and cold. For hot weather fo relaxes, 
 foftens, and weakens the main fpring, that it lofes 
 a great deal of its ftrength, which caufes the watch 
 to lofe time and go too flow. On the other hand, 
 cold frofty weather fo affeds the fpring, and it is 
 fo condenfed and hardened, that it becomes far 
 ftronger ; and by that means accelerates the mo- 
 tion of the v/atch, and makes her go fafter. The 
 difference of motion in a watch, thus occafioned 
 by heat and cold, will often amount to an hour, 
 and more in 24 hours. To remedy this, there is 
 a piece of machinery, called the Slide, placed near 
 the regulating fpring ; which being put forward or 
 backward, fliortens or lengthens the fpring, fo as 
 to make her keep time truly. 
 
 Some people have been fo filly as to think, that 
 the greater ilrength of a fpring arifes v/holly 
 
 fi'om
 
 112 A WATCH. 
 
 Fig. from its being made Ihorter, as this happens to be 
 one of the effeds of cold. But it is eafily demon- 
 
 ^7* llratcd that this is not the caufe. For let AB be 
 a Ipring as it is dilated by heat, and ab the fame 
 fpring contrafled by cold. Now if the fpring has 
 been contrafted in length, it mufl be proportional- 
 ly contradled in all dimenfions. Let /, />, d^ de- 
 note the length, breadth, and depth, in its cold, 
 and leaft dimenfions ; and r/, rb^ rd, the length, 
 breadth, and depth, in its hot and greateft dimen- 
 fions. Then (Prop. LIV.) the llrength of the 
 longer, to the ilrength of the fhorter, will be as 
 
 rb X rrdd hdd ^ , . . . , , , 
 J to -J- (confidering it weakened by the 
 
 length), and that is as rr to i, or as AB' to nV\ 
 So that the longer fpring, upon account of its be- 
 ing affedted with heat, is fo far from being weaker, 
 than the lliorter affeded with cold, that it is the 
 ftronger of the two. And therefore this difference 
 is not to be afcrib^d merely to the lengthning or 
 lliortning thereof-, but muft be owing to the na- 
 ture, texture and conftitution of the fteel, as it is 
 fome way or other affeded and changed by the 
 heat and cold. 
 
 And that there is fome change induced by the 
 cold, into the very texture of the mettal, is evi- 
 dent from this, that all forts of tools made of 
 iron or fteel, as fprings, knives, faws, nails, &c. 
 very eafily fnap and break in cold frofty weather, 
 which they will not do in hot weather. And that 
 property of fteel fprings is the true caufe, that thefe 
 forts of movements can never go true. 
 5(5. To make a calculation of the different forces re- 
 quifite to make a watch gain or lofe any number 
 of minutes, as fuppofe half an hour in 24 -, and I 
 have often experienced it to be more. By Cor. 4. 
 Prop. VI. the produd of the torce and Iquare of 
 
 the
 
 Sea. V. A WATCH. 113 
 
 the time, is as the product of the body and fpace Fig. 
 defcribed, which here is a given quantity. For 66, 
 the matter of the balance remains the fame in hot 
 as cold weather ; and fo does the length of the 
 fwing, which here is the fpace defcribed. There- 
 fore the force is reciprocally as the fquare of the 
 tim.e of vibrating) or direftly as the fquare of the 
 number of vibrations in 24 hours. Therefore the 
 force with the warm fpring, is to the force with the 
 cold one ; as the fquare of 23 i- hours, to the fquare 
 of 24-, that is, nearly as 23 to 24. So that if a 
 fpring was to contra6l half an inch in a foot in 
 length, without altering its other dimenfions, it 
 would but be fufficient to account for that phe- 
 nomenon ; but this is forty times more than the 
 lengthening and fhortening by heat and cold, for 
 that does not alter fo much as a thoufandth part, 
 as is plain from experiments. 
 
 The cafe being thus, a clock or watch going by 
 a fpring, can never be made to keep time truly, 
 except it be alv/ays kept to the fame degree of heat 
 or cold, which cannot be done without conftant 
 attendance. And if any ibr-t of mechanifm be con- 
 trived to corred this ; yet as fuch a thing can only 
 be made by guefs, it cannot be trufted to at fea, but 
 only for fbort voyages. But no motion however 
 regular, can ever anfwer at fea, where the irregu- 
 lar motion of the fliip will continually diilurb it ; 
 add to this, that the fmall compafs a watch is con- 
 tained in, makes it eafier difturbed, than a larger 
 machine would be •, but to fuppofe that any regu- 
 lar motion can fubfift among ten thoufand irregular 
 motions, and in ten thoufand different direcftions, 
 is a mod glaring abfurdity. And if any one with 
 fuch a machine would but make trial of it to the Eafl: 
 Indies, he would find the abfurdity and difappoint- 
 ment. And therefore I never expeft to fee fuch a 
 time keeper, or any fuch thing as a watch or clock 
 I going
 
 114 A DESCENDING CLOCK. 
 
 Fig. going by a fpring, to keep true time at fea. But 
 
 66. time will dilcover all things. 
 
 As to pendulum clocks, their irregularity in the 
 fame latitude is owing to nothing but the length^ 
 ning or Ihortning of the pendulum •, which is a 
 mere trifle to the other. But then they would be 
 infinitely more difturbed at fea, than a watch ; and 
 in a ftorm could not go at all. In different lati- 
 tudes too, another irregularity attends a pendulum, 
 depending on the different forces of gravity. Tho' 
 this amounts but to a fmall matter, yet it makes 
 a confiderable variation, in a great leng-th of time. 
 For in fouth latitudes, v/here the gravity is lefs, a 
 clock lofes time. And in north latitudes, where 
 the gravity is greater, it gains time. So that none 
 of thefe machines are fit to meafu;e time at fea, al- 
 tho' ten times ten thoufand pounds fliould be given 
 away for making them. 
 
 A DESCENDING CLOCK. 
 
 6S. Fig. 68. is a clock defcending dov/n an inclined 
 plane. This confifts of a train of watch work, 
 contained between tv/o circular plates AB, CD, 
 4 inches diameter, fixed together by a hoop an 
 inch'and half broad, inclofing all the work. The 
 inner work confiils of 5 wheels, the fame as in a 
 watch, only there is a fpur wheel inflead of the 
 contrate wheel, as 4 ^ ^ is the balance, v/hofe pa- 
 -lats play in the teeth of the crown wheel 5. Here 
 is no fpring to give it motion, but inilead thereof, 
 the weight W is fixed to the wheel i, and fo ad- 
 iufled for weight, that it may balance the lower 
 lide, and hinder it from rolling down the plane. 
 
 .Now Vvhilft the v/eight W moves the wheel i, this 
 wheel by moving about, caufes the weight W to 
 
 .defcend, by which it ceafes to be a balance for the 
 oppcfite fide, and therefore that fide begins to de- 
 fcend.
 
 Sea. V. A DESCENDING CLOCK. 115 
 
 fcend, till the weight W be raifed high enough Fig. 
 again to become a balance, which mull be about 68. 
 the pofition it appears in the figure. Thus wuilil 
 wheels move gradually about, the weight W de- 
 fcends gradually, which makes the body of the 
 machine turn giaduaiiy round, and defcend down 
 the inclined plane PQ; making one revolution in 
 12 hours. And therefore to have her to go 24 or 
 30 hours ; the length of the plane PQ^muft be 2 
 or 2 ' circumferences of the plates. Before the 
 weight W is fixed to the wheel i, fome lead or 
 brafs mult be foldered on the fide E oppofite to 
 the wheels 2, 3, 4, &c. for the v/heel 1 muik be 
 in the center. And then the lead or brafs muit 
 be filed away till the center of gravity of the ma- 
 chine be in the center of the plates. And to hin- 
 der the machine from fiiding, the edges of the 
 plates muil be lightly indented. The inclined 
 plane PQ may be a board, which muft be elevated 
 10 or 12 degrees, but that is to be found by trials; , 
 for if flie s;o too (low the end P muft be raifed ; 
 but if too fail it muft be lowered. When the 
 clock has gone the length of the board to Q, it 
 muft be let again at P. The fore fide CD is di- 
 vided into liours, and a pin is fixed in the center 
 at G, on which the hand FGH, always hangs 
 loofely in a perp. pofition, with the heavy end H 
 downward. And the end F ftiews the hour of 
 the day. So that the hours come to the hand, 
 and not the hand to the hours. 
 
 The board PQ muft be be perfeftly ftreight 
 from one end to the other ^ or elle ftie vv'ill go laf- 
 ter in fome places, and llov/er in others. 
 
 The circle with hours ought to be a narrow rim 
 of brafs, movable round about, by the help of 
 of one or more pins placed in it ; fo that it may 
 be fet to the true time. 
 
 I 2 The
 
 ii6 A DESCENDING CLOCK. 
 
 Fig. The weight W ferves for two ufes, i, to be a 
 63. counterpoife to the fide A; and 2, by its weight 
 to put the clock in motion. 
 
 The weight W muft be fo heavy as to make 
 the clock keep time, when it has a proper de- 
 gree of elevation as 45 degrees; and then the 
 board muft have an elevation of 10 or 12 de- 
 grees. If fhe go too faft, with thefe pofitions, 
 take fome thing off the weight j if too flow, add 
 fomething to it. 
 
 SEC T.
 
 TlJE/>a.2ip:
 
 PLTH/^.i 
 
 r
 
 [ 117 ] 
 
 Fig. 
 
 SECT. VI. 
 
 Hydrostatics and Pneumatics. 
 
 DEFINITION I. 
 
 /I Fluids is fuch a body v/hofe parts are eafily 
 moved among themfelves, and yield to any 
 force ading againft them. 
 
 DBF. II. 
 
 Hydrojiatics^ is a fcience that demonftrates the 
 properties of fluids. 
 
 D E F.. III. 
 
 Hydraulics^ is the art of railing water by engines. 
 
 "D E F. IV. 
 
 Pneumatics^ is that fcience which ihews the pro- 
 perties of the air. 
 
 D E F. V. 
 
 A fountain or jet d'eau, is an artificial fpout of 
 water. 
 
 PROP. LXIII. 
 
 If one part of a fluid he higher than another, the 
 higher parts will continually defcend to the lower places^ 
 and will not be at refl, till the furface of it is quite 
 level. 
 
 For the parts of a fluid being movable every 
 
 way, if any part is above the reft, it will defcend 
 
 by its own gravity as low as it can get. And af- 
 
 terv/ards other parts that are now become higher, 
 
 I 3 will
 
 ii8 HYDROSTATICS. 
 
 Ficr. will defcend as the other did, till at laft they will all 
 be reduced to a level or horizontal plane. 
 
 Cor. I . He7ice water that communicates ly means 
 cf a channel cr ptpe, with other water j will Jetlle 
 at the fame hight in both places. 
 
 Cor. 2. For the fi.me reafon^ if a fluid gravitates 
 towaraS a center •, it wiV. dijpofe it f elf into a fpherical 
 figure, whofe center is the center of force. As the 
 fca in refpe^l of the earth. 
 
 PROP. LXIV. 
 
 If a fuiid he at reft in a vefjcl whofe hafe is paral- 
 lel to the hot izon •, etiual parts cf the hafe are equally 
 preffcd hy the fluid. 
 
 For upon every part of the bafe there is an 
 equal column of the fiuid lupported by it. And 
 as all thc'.e columns are of equal weight, they muft 
 prefs the bale equally ; or equal parts of the bafe 
 will fuftain an equal prefilire. 
 
 Cor. 1 . All parts of the fluid prefs equally at the 
 fame depth. 
 
 For imagine a plain drawn thro' the fluid paral- 
 lel to the horizon. Then the preflure will be the 
 fame in any part of that plane, and therefore the 
 parts of the fiuid at tlie fame depth fuftain the 
 fame prefilire. 
 
 Cor. 2 . 1'he prcffure of a fluid at any depth, is as 
 the depth of the fluid. 
 
 For the prefilire is as the weight, and the weight 
 is as the hight of a column of the fluid. 
 
 PROP.
 
 Sea. VL HYDROSTATICS. n^ 
 
 PROP. LXV. ^^' 
 
 If a fluid is coinprejfed by its weight or otherwife ; 
 at any point it prejjes equally^ in all manner of dire^ions. 
 
 This arifes from the nature of fluidity, which 
 is, to yield to any force in any diredion. If it 
 cannot give way to any force applied, it will prefs 
 againfl other parts of the fluid in direction of that 
 force. And the preffure in all direftions with be 
 the fame. For if any one was lefs, the fluid 
 would move that way, till the preflfure be equal 
 every way. 
 
 Cor. In a7ty veffel containing a fluid ; the preffure 
 is the fame againfi the bottom^ as againfi the Jides^ 
 er even upwards^ at the fame depths 
 
 PROP. LXVI. 
 
 The preffure of a fluid upon the bafe of the con- r 
 tainting vejj'el, is as the bafe. and perpendicular alti^ ^* 
 tude ; zvhatever be the figure of the veffel that con- 
 tains it. 
 
 Let ABIC, EGKH be two vems. Then 
 (Prop. LXIV. Cor. 2.) the prefllire upon an inch 
 on the bafe AB = hight CD X i inch. And the 
 prefllire upon an inch on the bafe HK is iz hight 
 FH X I inch. But (Prop. LXIV.) equal parts of 
 the bafes are equally preflTed, therefore the preflijre 
 on the bafe AB is CD X number of inches in AB; 
 and preflfure on the bafe HK is FH X number of 
 inches in HK. That is, the prefllire on AB is to 
 the prefllire on LIK ; as bafe AB X hight CD, to 
 the bafe HK X hight FH. 
 
 Cor. I . Hence if the hight s be equal, the preffires 
 Are as the bafes. And if both the hights and bafes be 
 
 I 4 equal -^
 
 120 HYDROSTATICS. 
 
 Fig. equal j the pjrjfures are equal in both j thd' their con- 
 
 6c). tails he never fo different. 
 
 For the rcafon that the v*'ider vefiel EK, lias no 
 greater preHure at the bottom, is, becaufe the 
 oblique fides EH, GK, take off part of the weight. 
 And in the narrower veffel CB, the fides CA, IB, 
 re -act againft the preffure of the water, which is all 
 alike at the fame depth •, and by this re-a6Vion the 
 preffure is increafed at the bottom, fo as to be- 
 come the fame every where. 
 
 Cor. 2 . T'he preffure againjt the hnfe of any veffel^ 
 is the ff.rne as of a cylinder of an equal bafe and hight. 
 
 >jo. Cor. 3. If there be a recurve tube ABF, in which 
 ere tvjo different fluids CD, EF. 'Their hights in 
 the two legs CD, EF, i^ill be reciprocally as their 
 fpecific gravities, when they are at refl. 
 
 For if the fluid EF be twice or thrice as light 
 - as CD ; it mufl: have twice or thrice the hight, to 
 have an equal preffure, to counterbalance the other. , 
 
 PROP. LXVII. 
 
 *^ I , Jf a body of the fame fpecific gravity of a fluid j 
 be immerfcd in it, it will refl in any place of^ it. A 
 body of greater derjity will fink ; and one of a lefs 
 denfjy will fwim. 
 
 Let A, B, C be three bodies ; whereof A is 
 lighter bulk for bulk than the fluid ; B is equal •, and 
 C heavier. * The body B, being of the fame den- 
 fity, or equal in weight as fo much of the fluid ; 
 it will prefs the fluid under it jufl; as much as if 
 the fpace was flllcd with the fluid. The preffure 
 then will be the fame all around it, as if the fluid 
 Was there, and confequently there is no force to put 
 it out of its place. But if the body be lighter, 
 
 the
 
 Sea. VI. HYDROSTATICS. 121 
 
 the preffure of it downwards will be lefs than be- Fig. 
 fore; and lefs than in other places at the fame 71. 
 depth j and confequently the leffer force will give 
 way, and it will rife to the top. And if the body- 
 be heavier, the preffure downwards will be greater 
 than before ; and the greater preffure will prevail 
 and carry it to the bottom. 
 
 Cor. I . Hence if fever al bodies of different fpecific 
 gravity be immerfed in a fluid; the heavieji will get 
 the loweft. 
 
 For the heavieil are impelled with a greater 
 force, and therefore will go fafteft dov/n. 
 
 Cor. 2. A body immerfed in a fluids lofes as much 
 weighty as an equal quantity of the fluid weighs. And 
 the fluid gains it. 
 
 For if the body is of the fame fpecific gravity 
 as the fluid •, then it will lofe all its v/eight. And 
 if it be lighter or heavier, there remains only the 
 difference of the weights of the body and fluid, to 
 move the body. 
 
 Cor. 3. All bodies of equal magnitudes^ lofe equal 
 'weights in the fame fluid. And bodies of different 
 magnitudes lofe weights proportional to the magnitudes. 
 
 Cor. 4. The weights loft in different fluids, by im- 
 merging the fame body therein, are as the fpecific gra- 
 tuities of the fluids. And bodies of equal weight, lofe 
 weights in the fame fluid, reciprocally as the fpecific 
 gravities of the bodies. 
 
 Cor. 5. The weight of a body fwimming in a. 
 fluid, is equal to the weight of as much of the fiuidj 
 as the immerfed part of the body takes up. 
 
 For the preffure underneath the fvvimming body 
 is jull the fame as fo much of the immerfed fluid ; 
 and therefore the weights are the fame. 
 
 Cor*
 
 122 H Y D R O S T A T I C S. 
 
 Fig. Cor. 6. Hence a body will fink deeper in a lighter 
 
 7 1 . f.idd than in a heavier. 
 
 Cor. 7- Hence appears the reafon why we do not 
 feel the whole weight of an immerfed body, till it be 
 drawn quite out of the water. 
 
 PROP. LXVIII. 
 
 72. If a fluid runs thro* a pipe^ fo as to leave no vacui- 
 ties \ the velocity of the fluid in different parts of ity 
 will he reciprocally as the tranfverfe feHions, in thefe 
 parts. 
 
 Let AC, LB be the fedions at A and L. And 
 let the part of the fluid ACBL come to the place 
 acbl 1 hen will the folid ACBL = folid acbl-, 
 take away the part acWL common to both ; and 
 vv^e have ACca z: LB^/. But in equal folids the 
 bafes and hights are reciprocally proportional. But 
 if D/ be the axis of the pipe, the hights D^, F/, 
 pafied thro* in equal times, are as the velocities. 
 ■Therefore, fedion AC : feftion LB : : velocity 
 along F/ : velocity along D^. 
 
 PROP. LXIX. 
 
 yZ' Jf AD is a veffel of water or any other fluid \ B 
 a hole in the bottom or fide. Then if the veffel be al- 
 ways kept full; in the time a heavy body falls 
 thro' half the hight of the water above the hole AB, 
 a cylinder of water will flow cut of the hole, whofe 
 hight is AB, and bafe the area of the hole. 
 
 The prefTure of the water againft the hole B, 
 by which the motion is generated, is equal to the 
 weight of a column of water whofe hight is AB, 
 and bafe the area B (by Cor. 2. Prop. LXVL). But 
 equal forces generate equal motions j and fince a 
 
 cylinder
 
 Sea. VI. HYDROSTATICS- 12^ 
 
 cylinder of water falling thro' -[- AB by its gravity. Fig. 
 acquires fuch a motion, as to pafs thro' the whole 73^ 
 hight AB in that time. Therefore in that time 
 the water running out muft acquire the fame mo- 
 tion. And that the effluent water may have the 
 fame motion, a cylinder miuft run out whole length 
 is AB ; and then the fpace defcribed by the water 
 in that time will alfo be AB, for that fpace is the 
 length of the cylinder run out. Therefore this is 
 the quantity run out in that time. 
 
 Cor. I. The quantity run out in any time is equal 
 to a cylinder or prifin^ whofe length is the fpace de- 
 fcribtd in that time by the velocity acquired by fall- 
 ing thro* half the hight ^ and whofe bafe is the hole. 
 
 For the length of the cylinder is as the time of 
 running out. 
 
 Cor. 2. The velocity a little without the hole^ is 
 greater than in the hole ; and is nearly equal to the 
 velocity of a body falling thro"" the whole hight AB. 
 
 For without the hole the ftream is contradled by 
 the water's converging from all fides to the cen- 
 ter of the hole. And this makes the velocity- 
 greater in about the ratio of i to ^Z^. 
 
 Cor. 3. The water fpouts out with the fame velo^ 
 city, whether it be downwards, or fdeways, or up- 
 wards. And therefore if it be upwards, it afcends- 
 nearly to the hight of the water above the hole. 
 
 Cor. 4. The velocities and likewife the quantities of 
 the fpouting water, at different depths j will be as 
 the fquare roots of the depths. 
 
 Scholium. 
 
 From hence are derived the rules for the conftruc- 74. 
 tion of fountains or jets. Let ABC be a refer- 
 voir of water, CDE a pipe coming from it, to 
 
 bring
 
 J.4 HYDROSTATIC S. 
 
 Y'l-y. bring water to the fountain which fpoiits up at E, 
 
 y^, to the hight EF, near to the level of the refervoir 
 AB. In order to have a fountain in perfedion, the 
 pipe CD muft be wide, and covered with a thin 
 plate at E with a hole in it, not above the fifth or 
 iixth part of the diameter of the pipe CD. And 
 this pipe muft be curve having no angles. If the 
 rei'ervoir be 50 feet high, the diameter of the hole 
 at E may be an inch, and the diameter of the pipe 
 6 inclies. In general, the diameter of the hole E, 
 ought to be as the fquare root of the hight of the 
 relervoir. When the water runs thro' a great 
 length of pipe, the jet will not rife fo high. A 
 jet never rifes to the full hight of the refervoir ; in 
 a 5 feet jet it wants an inch, and it falls fliort by 
 lengths which are as the Iquares of the higlits ; and 
 fmaller jets Icfe more. No jet will rife 300 feet high. 
 
 ^- A fmall fountain is eafily made by taking a 
 ftrong bottle A, and filling it half full of water; 
 cement a tube Bl very dole in it, going near the 
 botLom of the bottle. Then blow in at the top B, 
 to comprcfs the air within ; and the water will 
 fpout out atB. If a fountain be placed in the funfhine 
 and made to play, it will fhew all the colours of 
 the rainbov/, if a black cloth be placed beyond it. 
 A jet goes higher if it is not exactly perpendi- 
 cular ; for then the upper part of the jet falls to 
 one fide without refifcing the column below. The 
 refiftance of the air will alio deftroy a deal of its 
 motion, and hinder it from rifing to the hight of 
 the refervoir. Aifo the friction of the tube of pipe 
 of conduft has a great Hiare in retarding the motion. 
 
 ^8. If there be an upright velfel as AF full of wa- 
 ter, and feveral holes be made in the fide as B, C, 
 D : then the diftances, the water will fpout, upon 
 the horizontal plane EL, will be as the fquare 
 roots of the redtangles of the fcgments, ABE, 
 ACE, and ADE. For the fpaces will be as the 
 
 velocities
 
 Seft. VI. HYDROSTATICS. 125 
 
 velocities and times. But (Cor. 4.) the velocity of Fig. 
 
 tlie water flowing out of B, will be as \/AB, and 7^*'' 
 the time of its moving (which is the fame as the 
 time of its fall) will be (by Prop. XIII.) as 
 \/BE-, therefore the diftance EH is as s/AB X BE ; 
 and the fpace EL as v/ACE. And hence if two 
 holes are made equidiflant from top and bottom, 
 they will projed the water to the fame diftance, 
 for if AB =z DE, then ABE = ADE, which 
 makes EH the fame for both, and hence alfo it 
 follows, that the projedlion from the middle point 
 C will be furthefl ; for ACE is the greateft reflan- 
 gle. Thefe are the proportions of the diilances ; 
 but for the abfolute diftances, it will be thus. The 
 velocity thro' any hole B, v/ill carry it thro' 2AB • 
 in the time of falling thro' AB -, then to find hov/ 
 far it will move in the time of falling thro' BE. 
 Since thefe times are as the fquare roots of the 
 
 hights, it will be, v^AB : 2AB : : ^/BE : EH zz 
 
 BE . 
 
 2 AB v/;^ = 2 v/ABE ; and fo the ipace EL zi: 
 
 2 v^ACE. It is plain, thefe curves are parabolas. 
 For the horizontal motion being uniform ; EH 
 
 will be as the time ; that is, as \/BE, or BE will 
 be as EH% which is the property of a parabola. 
 
 If there be a broad velTel ABDC full of water, 
 and the top AB fits exadtly into it; and if the 
 imall pipe FE of a great length be foldered clofe 
 into the top, and if water be poured into the top 
 of the pipe F, till it be full ; it will raife a great 
 weight laid upon the top, vvith the little quantity 
 of water contained in the pipe ; which weight will 
 be nearly equal to a column of the fluid, whof:: 
 bafe is the top AB ; and hight, that of the pipe 
 EF. For the prefliire of the water againfl: the top 
 AB, is equal, to the weight &f that column of w:.- 
 
 o.
 
 126 HYDROSTATICS. 
 
 Fig. ter, by Prop. LXV. and Cor. And Prop. LXVI. 
 
 76. Cor. 2. 
 
 But here the tube muft net be too fmall. For 
 in capillary tubes the attracflion of the glafs will 
 take off its gravity. If a very fmiU tube be im- 
 merfed with one end in a vefiel of water, the wa- 
 ter will rife in the tube above the furface of the 
 water -, and the higher, the fmaller the tube is. 
 But inquickfilver, it defcends in the tube below the 
 external furface, from the repulfion of the glafs. 
 
 77. To explain the operation of a fyphon, which is 
 a crooked pipe CDE, to drav/ liquors off. Set the 
 fyphon v/ith the ends C, E, upwards, and fill it 
 with water at the end E till it run out at C ; to 
 prevent it, clap the finger at C, and fill the other 
 end to the top, and ftop that with the finger. Then 
 keeping both ends ftopt, invert the fhortcr end C 
 into a velTel of v/ater AB, and take off the fingers, 
 and the water will run out at E, till it be as low as 
 C in the veffel ; provided the end E be always 
 lower than C. Since E is always below C, the 
 hight of the column of v/ater DE is greater than 
 that of CD, and therefore DE muft out weigh 
 CD and defcend, and CD will follow after, being 
 forced up by the preffure of the air, which adts 
 upon the furface of the water in the veffel AB. 
 
 The furface of the earth falls belov/ the horizontal 
 level only an inch in 620 yards ; and in other dif- 
 tancesthe defcentsareasthefquares of the diftances. 
 79- And to find the nature of the curve DCG, form- 
 ing the jet IDG. Let AK be the hight or top of 
 the rcfervoir HP, and fuppofe the ftream to afcend 
 without any friction, or refiftance. By the laws of 
 .falling bodies the velocity in any place B, will be 
 as v/AB. Put the femidiamctcr of the hole at 
 V — c/, and AD — b. Then fince the fame wa- 
 ter paffcs thro' the feftions at D and B ; therefore 
 (Prop. LXVIII.) the velocity will be reciprocally 
 
 as
 
 Seen. VI. HYDROSTATICS. ny 
 
 I I T^* 
 
 as the fedion ; whence \/h '- ~Tj '- '• i/AB : -^j-i ; ^' 
 
 ^h v/^B 
 
 therefore gpi n: , , •> and dd^yh zi BC^v^AB, 
 
 whence AB X BC+ zz hd^ -, which is a paraboliform 
 figure v/hofe afTymptote is AK, for the nature of 
 the cataradbic curve DCG. And if the fluid was 
 to defcend thro' a hole, as IC; it would form it- 
 felf into the iame figure GCD in defcending. 
 
 PROP. LXX. 
 
 I'he re/ijlance any body meets with in moving thrd' 
 a fluid is as the fqiiare of the velocity. 
 
 For if any body moves with twice the velocity 
 of another body equal to it, it will ftrike againil 
 twice as much of the fluid, and with twice the ve- 
 locity •, and therefore has four times the refifl:ance ; 
 for that will be as the matter and velocity. And 
 if it moves with thrice the velocity, it ftrikes againil 
 thrice as much of the fluid in the fame time, with 
 thrice the velocity, and therefore has nine times the 
 reflfliance. And fo on for all other velocities. 
 
 Cor. If a fir earn of water whofe diameter is given^ 
 flrike againft an objlacle at reft •, the force againji it 
 will he as the fquare of the velocity of the flream. 
 
 For the reafon is the fame ; fince with twice or 
 thrice the velocity ; twice or thrice as much of the 
 fluid impinges upon it, in the fame time. 
 
 PROP. LXXI. 
 
 T^he force of a ftream of water againfi any plane 
 chjiacle at reft, is equal to the weight of a column of 
 water^ whofe bafe is the feciion of the ftream ; and 
 hight^ the fpace defended thro' by a falling body^ to 
 acquire that velocity. 
 
 For let there be a refervoir whofe hight is that 
 fpace fallen thro'. Then the water (by Cor. 2. 
 
 Prop.
 
 128 HYDROSTATICS. 
 
 Fio-. Prop. LXIX.) flowing our at the bottom of the re- 
 fervatoiy, has the fame motion as the ftream •, but 
 this is generated by the weight of that column of 
 water, which is the force producing it. And that 
 fame motion is deftroyed by the obitacle, therefore 
 the force againft it is the very fame : for there is 
 required as much force to deftroy as to generate 
 any motion. 
 
 Cor. I'he force of a ftream of water flowing out 
 at a hole in the bottom of a reJervator)\ is equal to 
 the weight of a column of the flidd of the far,ie hight 
 and whofe bafe is the hole. 
 
 PROP. LXXII. Prob. 
 To find the fpecific gravity of folids or fluids. 
 
 1 . For a folid heavier thayi water. 
 Weigh the body feparately, firft out of water, 
 
 and then fufpended in water. And divide the 
 weight out of water by the difference of the weights, 
 gives the fpecific gravity j reckoning the fpecific 
 gravity of water i. 
 
 For the difference of the weights is equal to the 
 weight of as much water (by Cor. 2. Prop. LXVII.) ; 
 and the weights of equal magnitudes, are as the 
 fpecific gravities ; therefore the difference of thefe 
 weights •, is to the weight of the body ; as the fpe- 
 cific gravity of water 1, to the fpecific gravity of 
 the body. 
 
 2. For a body lighter than water. 
 
 Take a piece of any heavy body, fo big as be- 
 ing tied to the light body, it may fink it in water. 
 Weigh the heavy body in and out of water, and 
 find the lofs of weight. Alfo weigh the compound ' 
 both in and out of water, and find alfo the lofs of 
 
 weight.
 
 Sea. VI. HYDROSTATICS. 129 
 
 weight. Then divide the weight of the light bo Fig. 
 by (out of water), by the difference of thefe lofieSi 
 gives the fpecific gravity ; the fpecific gravity of 
 water being i. ' 
 
 For the laft: lofs is zi weight of water equal in 
 magnitude to the com- 
 pound. 
 And the firft lofs is zz weight of water equal in 
 magnitude, to the heavy 
 body. 
 Whence the dif. lofles is z: weight of water equal in 
 
 magnitude to the light 
 body, 
 and the weights of equal magnitudes, being as the 
 fpecific gravities ; therefore the difference of the 
 loiTes, (or the weight of water equal to the light 
 body) : weight of the light body : : fpecific gravi- 
 ty of water i : fpecific gravity of the light body. 
 
 3. For a fluid of any fvrt. 
 
 Take a piece of a body whofe fpecific gravity 
 you know ; weigh it both in and out of the 
 fluid ; take the difference of the weights, and mul- 
 tiply it by the fpecific gravity of the folid body, 
 and divide the product by the weight of the body 
 (out of water), for the fpecific gravity of the fluid. 
 
 For the difference of the weights in and out of 
 •water, is the weight of fo much of the fluid as 
 equals the magnitude of the body. And the weight 
 of equal magnitudes being as the fpecific gravities ; 
 therefore, weight of the folid : difference of the 
 weights (or the weight of fo much of the fluid) : : 
 fpecific gravity of the folid : io the fpecific gravity 
 of the fluid. 
 
 Example. 
 
 I weighed a piece of lead ore, which was 124 
 grains; and in water it weighed 104 grains, the 
 
 K diiferenc*
 
 ,30 HYDROSTATICS. 
 
 vity of the ore. 
 
 ^^^" difference is 20 i then ^ = 6.2 ; the fpecific gra- 
 
 A table of fpecific gravities. 
 
 Fine gold — . — . — 
 Standard gold — — — . 
 Quickfilver — — — 
 Lead — — . — 
 
 Fine filver — — "^ 
 
 Standard filver — — — 
 Copper — .— — . 
 
 Copper half-pence — — 
 Gun metal — . — — 
 
 Fine brafs — — . — 
 
 Caftbrafs — — — 
 
 Steel _ — — 
 
 Iron — "~ "~ 
 
 Pewter *— — — 
 
 Tin ^ — — 
 
 Caft iron .^ — — 
 
 Lead ore .^ — — 
 
 Copper ore •— — » — 
 
 Lapis calaminaris — — 
 
 Load Hone — . — — 
 
 Antimony — — . — 
 Diamond — — — 
 
 Ifland chriftal • — — — 
 Stone, hard — — . — 
 
 Rock chriftal — * — ~ 
 Glafs — I — — 
 
 Flint — — — 
 
 Common (lone — — 
 Chriftal — — — 
 
 Brick -- — ^ 
 
 Farth — ^_ .^ 
 
 Horn »- — --. 
 
 19.640 
 18.888 
 14.000 
 II. 340 
 11.092 
 10.536 
 
 9.000 
 
 S.915 
 
 8.784 
 
 8.350 
 
 8.100 
 
 7.850 
 
 7.644 
 
 7-47 J 
 7.320 
 7.000 
 6.200 
 5.167 
 5.000 
 
 4-930 
 4.000 
 
 2.720 
 
 2.700 
 2.650 
 2.600 
 2.570 
 . 2.500 
 2.210 
 2.000 
 1.984 
 Z.840 
 Ivory
 
 Sea.VI. HYDROSTATICS. i^t 
 
 Ivory -^ •— *— 1.820 Fig» 
 
 Chalk — « — — 1.793 
 
 AUum »— — — 1-714 
 
 Clay «— .— . •^ 1. 712 
 
 Oil of vitriol — , — — 1.700 
 
 Honey — — — 1.450 
 
 Lignum vitse — -^ — . 1.327 
 
 Treacle — — — 1.290 
 
 Pitch — • — — 1. 1 50 
 
 Rozin — — . — . 1. 100 
 
 Mohogany •— — — 1.065 
 
 Amber — — «-. 1.040 
 
 Urine — — — 1.032 
 
 Milk — — — 1.031 
 
 Brazil — — — 1.03 1 
 
 Box — — . — 1.030 
 
 Sea water — — . — . 1.030 
 
 Ale — — — 1.028 
 
 Vinegar — — — i.026- 
 
 Tar — — — 1.015 
 
 Common clear water — — — i.ooo 
 
 Bee wax — — — .^^^ 
 
 Butter — — , — .940 
 Linfeed oil — — — .^sz 
 Brandy — — — .927 
 
 Sallad oil — — — • .913 
 
 Logwood — — — *9i3 
 
 Ice — — — .908 
 
 Oak — — — .830 
 
 Afh — — — ,S^o 
 
 Elm — — — .820 
 
 Oil of turpentine ^ — -i— — . .810 
 
 Walnut tree — « — , — .650 
 
 Fir — — . — .580 
 
 Cork — . — — . .238 
 Ne\r fallen fnow — — — • .08^ 
 
 Air .^ — , — .ooia 
 
 K 2 Cor;
 
 1^2 HYDROSTATICS. 
 
 Fig. Cor. I. As the weight loft in a fluid, is to the ab- 
 folute weight of the body •, fo is the fpecific gravity of 
 the fluid, to the fpecific gravity of the body. 
 
 Cor. 2. Having the fpecific gravity of a body^ 
 and the wei.^ht of it ; the folidity may be found thus ; 
 multiply the weight in pounds by 62 4-' "^hey fay as 
 that produS to 1 \ fo is the weight of the body in 
 ^pounds, to the content in feet. And having the con- 
 tent given, one may find the weight, by working 
 backwards. 
 
 For a cubic foot of water weighs 62 \ lb. aver- 
 dupoife \ and therefore a cubic foot of the body 
 weighs 62 '- X by the fpecific gravity of the body. 
 "Whence the weight of the body, divided by that 
 produ(5t, gives the number of feet in it. Or as i, 
 to that product j fo is the content, to the weight. 
 
 Scholium. 
 Iq^ The fpecific gravities of bodies may be found 
 with a pair of fcales 5 fufpending the body in wa- 
 ter, by a horfe hair. But there is an inftrument 
 for this purpofe called the Hydrofiatical Balance., 
 the conftrudtion of which is thus. AB is the ftand 
 and pedeftal, having at the top two cheeks of 
 tlcel, on which the beam CD is fufpended, which 
 is like the beam of a pair of fcales, and muft play 
 freely, and be it felf exactly in equilibrio. To 
 this belongs the glafs bubble G, and the glafs 
 bucket H, and four other parts E, P\ I, L. To 
 thele are loops faftened to hang them by. And 
 the weights of all thefe are lb adjufted, that E =z 
 F 4- the bubble in water, or n: I + the bucket 
 out of water, or — I + L + the bucket in water. 
 Vv^hence L zi difference of the weights of the 
 bucket in and out of water. And if you pleafc 
 you may have a weight K, fo tliat K 4- bubble in 
 water — bubble out of water i or elie find it in 
 
 grains.
 
 Sea. VI. HYDROSTATICS. 133 
 
 grains. The piece L has a flit in it to flip it upon Fig. 
 the fliank of I. 80. 
 
 It is plain the weight K zi weight of water as big 
 as the bubble, or a water bubble. 
 
 Then to find the fpecific gravity of a folid. 
 
 Hang E at one end of the balance, and I and 
 the bucket with the folid in it, at the other end ; 
 and find what weight is a balance to it. 
 
 Then flip L upon I, and immerge the bucket 
 and folid in the water, and find again what weight 
 balances it. Then the firfl; weight divided by the 
 diff'erence of the weights, is the fpecific gravity of 
 the body ; that of water being i . 
 
 For fluids. ^ 
 
 Hang E at one end, and F with the bubble at 
 the other ; plunge the bubble into the fluid in the 
 veflel MN. Then find the weight P which makes 
 a balance. Then the fpecific gravity of the fluid 
 
 K + P , „. , ., K — P 
 
 is z: — Y^ — ' when P is laid on F j or zz — ^^ — > 
 
 when P is laid on E. 
 
 For E being equal to I + the bucket •, the firfl: 
 weight found for a balance, is the weight of the ' 
 folid. Again, E being equal to I + L + the 
 bucket in water •, the weight to balance that, is 
 the weight of the folid in water ; and the difference, 
 is ~ to the weight of as much water. Therefore 
 (Cor. 1.) the firfl; weight divided by that difl'erence, 
 is the fpecific gravity of the body. 
 ► Again, fince E is iz to F -f- the bubble in wa- 
 ter j therefore P is the difference of the weights, 
 of the fluid and fo much water •, that is, P z: dif- 
 ference of K and a fluid bubble j or P z: fluid — K, 
 when the fluid is heavier than water, or when P is 
 laid on F. And therefore P z: K — the fluid 
 K 3 bubble.
 
 t54. PNEUMATICS. 
 
 Fig. bubble, when contrary. Whence the fluid bubble 
 
 80. ir: K + P, for a heavier or lighter fluid. And the 
 
 fpecific gravities being as the weights of thefe equal 
 
 bubbles ; fpecific gravity of water : fpecific gravi- 
 
 K + P 
 ty of the fluid : : K : K -tP : : i : — ^ the fpe- 
 cific gravity of the fluid. Where if P be o, it is 
 the fame as that of water. 
 
 PROP. LXXIII. 
 
 ^he air is a heavy body, and gravitates on all parts 
 cf the furface of the earth. 
 
 That the air is a fluid is very plain, as it yields 
 to any the leafl: force that is imprefled upon it, 
 without making any fenfible refiftance. But if it; 
 be moved brifkly, by fome very thin and light bo- 
 dy, as a fan, or by a pair of bellows, we become 
 very fenfible of its motion againft our hands or 
 face, and likewife by its impelling or blowing away 
 any light bodies, that lie in the way of its motion. 
 Therefore the air being capable of moving other 
 bodies by its impulfe, mufl: it felf be a body ; and 
 muft therefore be heavy like all other bodies, in 
 proportion to the matter it contains \ and will con- 
 sequently prefs upon all bodies placed under it. 
 And being a fluid, it will dilate and fpread itfelf 
 all over upon the earth : and like other fluids will 
 gravitate upon, and prefs every where upon its fur- 
 face. The gravity and preflure of the air is alfo 
 evident from experiments. For (fig. 70.) if water, 
 &c. be put into the tube ABF, and. the air be 
 drawn out of the end F by an air-pump, the water 
 will afcend in the end F, and defcend in the end 
 A, by reafon of the prefilire at A, which was 
 taken off^ or diminiflied at F. There are number- 
 lefs experiments of th's fort. And tho' thele pro- 
 perties
 
 VlJM.^aj34
 
 . IL-A.,,.. 
 
 pi.Tnr.^tfjj-/. 
 
 r
 
 Sea. VI. P N E U M A T I C S. 135 
 
 perties and efFefts are certain, yet the air is a fluid Fig. 
 fo very fine and fubtle, as to be perfectly tranfpa- 70. 
 rent, and quite invifible to the eye. 
 
 Cor. I. ^he air, like other fluids, will, hy its 
 weight and fluidity, infinuate itfelf into all the cavi- 
 ties, and corners within the earth; and there prefs 
 with fo much greater force, as the places are deeper. 
 
 Cor. 2. Hence the atmofphere, or the whole hody of 
 air furrounding the earth, gravitates upon the fur- 
 faces of all other bodies, whether folid or fluid, and 
 that with a force proportional to its weight or quan- 
 tity of matter. 
 
 For this property it mufl have in common with 
 all other fluids. 
 
 Cor. 3. Hence the preffure, at any depth of water, 
 or other fluid, will he equal to the preffure of the fluid 
 Jogether with the prejfure of the atmofphere. 
 
 Cor. 4. Likewife all bodies, near the fur face of the 
 earth, lofe fo much of their weight, as the fame bulk of 
 fo much air weight. And confequently, they are fomething 
 lighter than they would he in a vacuum. But being 
 fo very fmall it is commonly neglected ; tho' in flri^- 
 nefs, the true or abfolute weight is the weight in vacuo, 
 
 PROP. LXXIV. 
 
 The air is an elaftic fluid, or fuch a one, as is ca- 
 pable of being condenfed or expanded. And it obferves 
 this law, that its denfity is proportional to the flora 
 that compreffes it, 
 
 Thefe properties of the air, are proved by e?:- 
 
 periments, of which there are innumerable. If 
 
 you take a fyringe, and thrufl the handle inwards, 
 
 you'll itd the included air a6c fl:rongly againfl: your 
 
 K 4 hand J
 
 136 PNEUMATICS. 
 
 Fig. hand -, and the more you thruft, the further the 
 pifton goes in, but the more it refifls j and taking 
 away your hand, the handle returns back to where 
 it was at firft. This proves its elafticityj and alfo 
 that air may be driven into a lefs fpace, and con- 
 denfed. 
 ^5. Again, take a ftrong bottle, and fill it half full 
 of water, and cement a pipe BI, clofe in it, going 
 near the bottom ; then injedl air into the bottle 
 thro' the pipe BI. Then the water will fpout out 
 at B, and form a jeti which proves, that the air i$ 
 firft condenfed, and then by its fpring drives out 
 the v/ater, till it become of the fam.e denfity as at 
 firft, and then the fpouting ccafcs. 
 
 81. Likewife if a veffcl of glafs AB be filled with 
 water in the vefTel CD, and then drawn up with 
 the bottom upwards ; if any air is left in the top 
 at A, the higher you pull it up, the more 'it ex- 
 pands ; and the further the glafs is thruft down in- 
 to the veflel CD, the more the air is condenfed. 
 
 82. Again, take a crooked glafs tube ABD open at 
 the end A, and clofe at D •, pour in mercury to 
 the hight BC, but no higher, and then the air in 
 DC is in the fame ftate as the external air. Then 
 pour in more mercury at A, and obferve where it 
 rifes to in both legs, as to G and H. Then you 
 may always fee that the higher the mercury is in 
 the leg BH, the lefs the fpace GD is, into which 
 the air is driven. And if the hight of the mercu- 
 ry FH be fuch as to equal the preflure of the at- 
 mofphere, then DG will be half DC -, if it be 
 twice the preflure of the atmofphere, DG will be 
 4 DC, &c. So that the denfity is always as the 
 weight or comprefllon. And here the part CD is 
 fuppofed to be cylindrical. 
 
 Cor. I . The fpace that any quantity of air takes iip^ 
 , is reciprocally as the force that comprcfj'cs it. 
 
 Cor.
 
 Sea. VI. PNEUMATICS. 137 
 
 Cor. 2. Jil the air near the earth is in a ftate o/Fio-, 
 compeJfiQn^ by the weight of the incumbent atmofphere. 
 
 Cor. 3. 'The air is denfer near the earth, or at the 
 foot of a mountain^ than at the top cf it, and in 
 high places. And the higher from the earth the mors 
 rare it is. 
 
 Cor. 4, The fpring or elafitcity of the air is equal 
 to the weight of the atmofphere above it •, and pro- 
 duces the fame effetis, 
 ■ For they always balance and fuilain each other. 
 
 Cor. 5. Hevce if the denfity of the air be increafed ; 
 its fpring or elajticity will likewife be increafed in the 
 fame proportion. 
 
 Cor. 6. From the gravity and prefftire of the at- 
 mofphere, upon the fur faces of fluids, the fluids are 
 made to rife in any pipes or veffels, when the prejfure 
 within is taken off. 
 
 PROP. LXXV. 
 
 The expanfion and elafticity of the air is increafed 
 by heat, and decreafed by cold. Or heat expands, and 
 cold condenfes the air. 
 
 This is alfo matter of experience ; for tie a blad- 
 der very dole with fome air in it, and lay it before 
 the fire, and it will vifibly diftend the bladder ; and 
 burft it if the heat is continued, andencreafed high 
 enough/ 
 
 If a glafs veffel AB (Fig. 81.) with water in it, gi; 
 be turned upfide down, with a little air in the top 
 A ; and be placed in a veffel of water, and hung 
 over the fire, and any weight laid upon it to keep 
 it down ; as the water warms, the air in the top A, 
 will by degrees expand, till it fills the glafs, and 
 
 by
 
 13? PNEUMATICS. 
 
 Fig. by its ekftic force, drive all the water out of the 
 Si, glals, and a good part of the air will follow, by 
 continuing the veflel there. Many more experi- 
 ments may be produced proving the fame thing. 
 
 PROP. LXXVL 
 
 The air will prefs upon the furfaces of all fluids, 
 with any force j withqut puffing thro^ them, or enter- 
 inz into them. 
 
 If this was not fo, no machine, whofe ufe or 
 action depends upon the prelTure of the atmofphere, 
 could do its bufinefs. Thus the weight of the at- 
 mofphere preffes upon the furface of water, and 
 forces it up into the barrel of a pump, without any 
 air getting in, which would fpoil its working. Like- 
 vrife the preflure of the atmofjphere ke^ps mercu- 
 ry fufpended at fuch a hight, that its weight is equal 
 to that prefiure •, and yet it never forces itfelf thro' 
 the mercuiy into the vacuum above, though it 
 fiand never fo long. And whatever be the texture 
 or conftitution of that fubtle invifible fluid we call 
 Wr, yet it is never found to pafs through any fluid, 
 Vho' it be made to prefs never fo ftrongly upon it. 
 For tho' there be fome air inclofed in the pores of 
 almoft all bodies, whether folid or fluid ; yet the 
 •particles of air cannot by any force be made to pafs 
 'thro' the body of any fluid -, or forced through the 
 pores of it, although that force or prefllire be con- 
 ^tinued never fo long. And this feems to argue 
 that the particles of air are greater than the parti- 
 fcles or pores of other fluids •, or at leafl are of a 
 ^ruifture quite different from any of them. 
 
 PROP. 
 
 !
 
 I 
 
 Sea. VI. JPNEUMATIC S. i^^ 
 
 PROP. LXXVII. ^^^' 
 
 The weight or prejfure of the atmofphere^ upon any 
 hafe at the earth's furface ; is equal to the weight of 
 a, column of mercury of the fame bafe^ and whofe 
 hight is from 28 to 31 inches^ feldom more or lefs» 
 
 This is evident from the barometer, an inflrit- 
 ment which fhews the prelTure of the air ; which 
 at fome feafons Hands at a hight of 28 inches, fome- 
 times at 29, and 30, or 31. The reafon of this is 
 not, becaufe there is at fome times more air in the 
 atmofphere, than at others ; but becaufe the air 
 being an extremely fubtle and elaftic fluid, capable 
 of being moved by any impreffions, and many 
 miles high •, it is much difturbed by winds, and by 
 heat and cold ; and being often in a tumultuous 
 •agitation; it happens to be accumulated in fome 
 places, and confequently deprefled in other? ; by 
 which means it becomes denier and heavier where 
 jt is higher, fo as to raife the column of mercury 
 to 30 or 31 inches. And where it is lower, itjs 
 •rarer and lighter, fo as only to raife it to 28 or 29 
 inches. ,And experience fhews, that it feldom goes 
 without the limits of 28 and 31. 
 
 Cor. I . 'The air in the fame place does not always 
 continue of the fame weight ; but is fometimes heavier^ 
 4nd fometiines lighter ; but the mean weight of the at^ 
 mofphere, is that when the quickfilver fiands at about 
 ^94 inches. 
 
 Cor. 2. Hence the prefjure of the atmofphere upon 
 M fquare inch at the earth's furface^ at a medium^ is 
 very near 1 5 pounds^ averdupoife. 
 
 For an inch of quickfilver weighs 8.102 ounces. 
 
 Cor. 3. Hence alfo the weight orpreffure of the at' 
 mofphere^ in its lightefi and heavieft Jiate^ is equal to 
 
 $h&
 
 r4C PNEUMATICS. 
 
 Fig. tf^e weight of a column of laater, ^2 or ^6 feet bigh^ 
 or at a jnedium 34 feet. 
 
 For water and quickfilver are in weight nearly as 
 I to 14. 
 
 Cor. 4. If the air was of the fame denftty to the 
 top cf the atmofphere^ as it is at the earth ; its bight 
 would be about 5 \- miles at a medium. 
 
 For the weight of air and water are nearly as 
 12 to 1000. 
 
 Cor. 5. I^he denftty of the air in two places difiant 
 from each ether but a few miles, on the earth's fur- 
 face and in the fame level; may be looked on to be 
 the fame^ at the fame time. 
 
 Cor. 6. The denftty of the air at two different al- 
 titudes in the fame place, differing only by a few 
 jeet 5 nay be looked on as the fame. 
 
 Cor. 7. Jf the perpendicular hight of the top of a 
 fyphon from the water, be more than 34 feet, at a 
 fnean denftty of the air. 'The fyphon cannot be made 
 to run. 
 
 For the weight of the water in the legs will be 
 'greater than the prefTure of the atmofphcre, and 
 ■ both columns will run down, till they be 34 feet 
 high. 
 
 Cor. 8., Hence alfo the quickfilver rifes higher in the 
 barometer, at the bottom cf a mountain than at the 
 top. And at the bottom of a coal pit^ than at the 
 top of it. 
 
 S CHO-
 
 Sea. Vf. PNEUMATICS. 141 
 
 Scholium. Fig. 
 
 Hence the denfity of the air may be found at 
 any hight from the earth, as in the following table. 
 
 Miles 
 
 denfity 
 
 Miles 
 
 denfity 
 
 +" 
 
 .9564 
 
 10 
 
 .1700 
 
 ■a" 
 
 .9146 
 
 20 
 
 ,02917 
 
 3 
 
 .8748 
 
 30 
 
 .005048 
 
 I 
 
 .8372 
 
 40 
 
 .000881 
 
 2 
 
 .7012 
 
 50 
 
 .000155 
 
 3 
 
 .5871 
 
 100 
 
 .0000000298 
 
 4 
 
 .4917 
 
 
 
 5 
 
 .4119 
 
 
 
 The firft and third columns are the hight in 
 miles from the furface of the earth. And the fe- 
 cond and fourth columns, lliew the denfity at that 
 hight ; fuppofing the denfity at the furface of the 
 earth, to be i. 
 
 The denfity at any hight is eafily calculated by 
 this feries. Put r =z radius of the earth, h ir 
 hight from the furface, both in feet. Then the 
 denfity at the hight b, is the number belo:nging to 
 
 the logarithm, denoted by this feries — -7^ — ■ 
 
 h h h 
 
 — A — — B — — C &c. where A, B, C, &c. 
 
 are the preceding terms. The terms here will be 
 alternately negative and affirmative. But the firft 
 term alone is fufficient when the hight is but a few 
 miles. 
 
 By the weight and prefliire of the atmofphere, the 
 operations of pneumatic engines may be accounted 
 for and explained. I Ihall juft mention one or two. 
 
 A PUMP. 
 
 Fig. 83. is a common pump. AB the barrel or 
 body of the pump, being a hollow cylinder, made 
 
 of 
 
 83'
 
 142 PNEUMATIC?. 
 
 Fig. of wood or lead. CD the Iiandle movable about 
 83. the pin E. DF an iron rod moving about a pin 
 D ; this rod is hooked to the bucket or fucker FG, 
 which moves up and down within the pump. The 
 bucket FG is hollow, and has a valve or clack L 
 at the top opening upwards. H a plug f^xed at 
 the bottom of the barrel, being like wife hollow, 
 and a valve at I opening alfo upwards. BK the 
 bottom going into the well at K ; the pipe below 
 B need not be large, being only to convey the wa- 
 ter out of the well into tlie body of the pump^ 
 The plug H muft be fixed clofe that no water can 
 get between it and the barrel -, and the fucker FG, 
 is to be armed with leather, to fit clofe that no air or 
 water can get thro' between it and tiie barrel. 
 
 "When the pump is firft wrought, or any time in 
 dry weather when the water above the fucker 15 
 wafted, it muft be primed, by pouring in fome wa- 
 ter at the top A to cover the fucker, that no air get 
 through. Then raifmg the end C of the handle, 
 the bucket F defcends, and the water will rife thro' 
 the hollow GL, preffing open the valve L. Then 
 putting dov/n the end C raifes the bucket F, and 
 the valve L ftuits .by the weight of the water above 
 it. And at the fame time the preffure of the at- 
 mofphere forces the water up thro' the pipe KB, 
 and opening the valve i,'it paftes thro' the plug in- 
 to the body of the pump. A.nd when the fucker 
 G defcends again, the valve I lliuts, and the water 
 cannot return, but opening the v?.lve L, pafles thro* 
 the fucker GL. And when the lacker is raifed 
 again, the valve L fhuts again, and the water is 
 raifed in the pump. So that by the motion of the 
 pifton up and down, and the alternate opening and 
 fhutting of the two valves •, water is continually 
 raifed into the body of the pump, and difcharged 
 at the fpout M. 
 
 The
 
 Sea. Vl. PNEUMATICS. 145 
 
 The diftance KG, from the well to the bucket. Fig. 
 muft not be above 32 feet ; for the preflure of the 83, 
 atmofphere will raife the water no higher, and if 
 it is more, the pump will not work. It is evident 
 a pump will work better when the atmofphere is 
 heavy than when it is light, there being a twelfth 
 or fifteenth part difference, at different times. An^ 
 "when it is lighteft it is only equal to 32 feet. Where- 
 fore the plug H muft always be placed fo low, ^ ' 
 that the fucker GL may be within that compafs, 
 
 A BAROMETER. 
 
 Fig. 84. is a Barometer^ or an inftrument to mea- 84, 
 fure the weight of the air. It confiils of a glafs 
 cone ABC hollow within, filled full of mercury, 
 and hermetically fealed at the end C, {q that no 
 air be left in it. When it is fet upright, the mercury 
 defcends down the tube BC, into the bubble A, 
 which has a little opening at the top A, that the 
 air may have free ingrefs and egrefs. At the top 
 of the tube C, there muft be a perfed: vacuum. 
 This is fixed in a frame, and hung perpendicular 
 againft a wall. Near the top C, on the frame, is 
 placed a fcale of inches, fhewing how high the 
 mercury is in the t«be BC, above the level of it 
 in the bubble A, which is generally from 28 to 31 
 inches, but moftly about 29 or 30. Along with 
 the fcale of inches, there is alfo placed a fcale of 
 fuch weather as has been obferved to anfwer the fe- 
 veral hights of the quickfilver. Such a fcale you 
 have annexed to the 84th figure. In dividing the 
 fcale of inches, care muft be taken to make proper 
 allowance for the rifing or falling of the quickfilver 
 in the bubble A, which ought to be about half 
 full, when it ftands at 294^, which is the mean 
 hight. For whilft the quickfilver rifes an inch at 
 C, it defcends f little in the bubble A, and that 
 
 defcent
 
 144 PNEUMATICS. 
 
 Fig. delcent muft be deduced, which makes the divi- 
 84. fions be fomething lefs than an inch. Thele inches 
 muft be divided into tenth parts, for the more exad 
 meafuring the weight of the atmofphere. For the 
 pillar of mercury in the tube is always equal to the 
 weight of a pillar of the atmofphere of the fame 
 thicknefs. And as the hight of the quickfilver 
 increafes or decreafes, the weight of the air in- 
 creafes or decreafes accordingly. The tube muft 
 be near 3 feet long, and the bore not lefs than 4- 
 or -^ of an inch, in diameter, or elfe the quickfil- 
 ver will not move freely in it. 
 
 By help of the barometer, the hight of moun- 
 tains may be meafured by the following table. In 
 which the firft column is the hight of the moun- 
 tain, &c. in feet or miles •, the fecond the hight 
 of the quickfilver •, and the third the defcent of the 
 quickfilver in the barometer ; and this at a mean 
 denfity of the air. 
 
 Feet
 
 Sea. VI. PNEUMATICS. 
 
 Feet 
 
 High Barom. 
 
 Defcent 
 
 Feet 
 
 High Barom. 
 
 Defcent. 
 
 O 
 
 29.500 
 
 
 
 
 lOO 
 
 29.400 
 
 .100 
 
 2600 
 
 27.028 
 
 2.472 
 
 200 
 
 29.301 
 
 .199 
 
 2700 
 
 26.938 
 
 2.562 
 
 300 
 
 29.203 
 
 •^97 
 
 2800 
 
 26.848 
 
 2.652 
 
 400 
 
 29.105 
 
 '395 
 
 2900 
 
 26.758 
 
 2.742 
 
 500 
 
 29.007 
 
 '^93 
 
 3000 
 
 26.668 
 
 2.832 
 
 600 
 
 28.910 
 
 .590 
 
 3100 
 
 26.578 
 
 2.922 
 
 700 
 
 28 812 
 
 .688 
 
 3200 
 
 26.489 
 
 3.01 I 
 
 800 
 
 28.716 
 
 .784: 
 
 3300 
 
 26.400 
 
 3.100 
 
 900 
 
 28.619 
 
 .8S1. 
 
 3400 
 
 26.311 
 
 3.189 
 
 1000 
 
 28.523 
 
 •977 
 
 3500 
 
 26.222 
 
 3.278 
 
 1 100 
 
 28.428 
 
 1.072 
 
 3600 
 
 26.136 
 
 3-364 
 
 1200 
 
 28.332 
 
 1. 168 
 
 3700 
 
 26.049 
 
 3.451 
 
 1300 
 
 28.237 
 
 1.263 
 
 3800 
 
 25.961 
 
 3-539 
 
 1400 
 
 28.143 
 
 ^'357 
 
 3900 
 
 25.874 
 
 3.626 
 
 1500 
 
 28.048 
 
 1.452 
 
 4000 
 
 25.786 
 
 3-7'4 
 
 1600 
 
 27.954 
 
 1.546 
 
 4100 
 
 25.699 
 
 3.801 
 
 1700 
 
 27.860 
 
 1.640 
 
 4200 
 
 25.613 
 
 3.887 
 
 1800 
 
 2j.y66 
 
 1-734 
 
 4300 
 
 25.527 
 
 3-973 
 
 1900 
 
 27.672 
 
 1.82S 
 
 4400 
 
 25.441 
 
 4-059 
 
 2000 
 
 '^7-579 
 
 1.921 
 
 4500 
 4600 
 
 25-355 
 
 4-145 
 
 2100 
 
 27.487 
 
 2.013 
 
 25.270 
 
 4.230 
 
 2200 
 
 27-394 
 
 2.106 
 
 4700 
 
 25.185 
 
 4.315 
 
 2300 
 
 27.302 
 
 2.198 
 
 4800 
 
 25.101 
 
 4.399 
 
 2400 
 
 27.210 
 
 2.290 4900 
 
 25.017 
 
 4.483 
 
 2500 
 
 27.119 
 
 2.381 ' 5or>o 
 
 24.933 
 
 4.567 
 
 145 
 
 Fig. 
 
 TJ^e
 
 146 
 
 Fig. 
 84. 
 
 PNEUMATICS. 
 
 The Table contmued in Miles. 
 
 Miles 
 0. 
 
 H. Baiom. 
 
 Del'ceiU 
 
 Miles 
 
 H. Barom. 
 
 Delcent \ 
 
 29.50 
 
 
 
 
 0.25 
 
 28.21 
 
 1.29 
 
 3-25 
 
 16.57 
 
 12.93 
 
 0.50 
 
 26.98 
 
 2.52 
 
 3-50 
 
 15-85 
 
 13.65 
 
 0'75 
 
 25.80 
 
 370 
 
 i-75 
 
 15.16 
 
 14-34 
 
 I- 
 
 24.70 
 
 480 
 
 4- 
 
 14.50 
 
 15.00 
 
 1.25 
 
 23.62 
 
 5.88 
 
 4-25 
 
 13-87 
 
 15-63 
 
 1.50 
 
 22.60 
 
 6.90 
 
 4.50 
 
 13.27 
 
 16.23 
 
 ^-75 
 
 21.62 
 
 7.88 
 
 4-75 
 
 12.70 
 
 16.80 
 
 2. 
 2.25 
 
 2068 
 
 8.82 
 
 5- 
 
 12.15 
 
 ^7-35 
 
 19.78 
 
 9.721 
 
 5-25 
 
 11.62 
 
 17.88 
 
 2. 50 
 
 18.93 
 
 ^o.57\ 
 
 5'5o 
 
 II. 12 
 
 18.38 
 
 ^'75 
 
 18. II 
 
 ^^•39] 
 
 5-75 
 
 10.64 
 
 18.86 
 
 3- 
 
 17.32 
 
 12.18! 
 
 6. 
 
 lO.iB 
 
 19.32 
 
 This table is made from a table of the air's den- 
 fity, made as in Schol. Prop. LXXVII. And 
 then mulriplying all the numbers thereof by 29.5 
 the mean denfity of the air. For the denfity of 
 the air at any height above the earth is as the weight 
 of the atmofphere above it, (by Prop. LXXIV.) ; 
 and that is as the height of the mercury in the ba- 
 rometer. 
 
 A WATER BAROMETER. 
 
 A barometer may alfo be made of water as in 
 fig. 85, which is a water barometer. AB is a 
 glafs tube open at both ends, and cemented clofe 
 in the mourh of the bottle EF, and reaching very 
 •near the bottom. Then warming the bottle at the 
 fire, part of the air will fly out; then the' end A 
 is put into a vefTel of water mixed with cochineal, 
 which will go thro' the pipe into the bottle as it 
 grows cold. Then it is fet upright \ and the water 
 
 may
 
 Sea. VI. PNEUMATICS. 147 
 
 may be made to Hand at any point C, by fucking Ficr. 
 or blowing at A. And if this barometer be kept 85, 
 to the fame degree of heat, by putting it in a vef- 
 fel of fand, it will be very corrt6t for taking fmall 
 altitudes •, for a little alteration in the weight of 
 the atmofphere, will make the water at C rife or 
 fall in the tube very fenfibly. But if it be fuffer- 
 ed to grow warmer, the water will rife too high in 
 the tube, and fpoil the ufe of it j fo that it muft 
 be kept to the fame temper. 
 
 If a barometer was to be made of water put 
 into an exhaufted tube, after the manner of quick- 
 filver ; it v^ould require a tube 36 feet long or 
 more-, which could hardly find room withindoors. 
 But then it would go 14 times more exaft than 
 quickfilver -, becaufe for every inch the quickfilver 
 rifes, the water would rife 1 4 ; from whence every 
 minute change in the atmofphere would be dif- 
 cernable. 
 
 And the water barometer above defcribed will 
 fhew the variation of the air's gravity as minutely 
 as the other, if the bottle be large to hold a great 
 quantity of air. And in any cafe, by reducing 
 the bottle (fo far as the air is contained) to a cylin- 
 der ; and put D zr diameter of the bottle, d zz 
 diameter of the pipe, p 1= height of air, x n rif- 
 ing in the pipe, all in inches. Then the height of a 
 
 hill in feet will be nearly 1 + ~ X 7ix And 
 
 ^ pDD ' 
 
 if _y =z height of the hill or any afcent, Q — 
 
 ■ T->T-» • Then x rr - very near, at a 
 
 mean denfity of the air. 
 
 A THERMOMETER. 
 Fig. 86, is a thermometer^ or an inftrumcnt to gr 
 meafure the degrees of heat and cold. AB is a 
 
 hollovs(
 
 148 PNEUMATIC?;. 
 
 Fisf. hollow tube near two foot long, with a ball at the 
 86. bottom ', it is filled with fpirits of wine mixed with 
 cochineal, half way up the neck j which done, it is 
 heated very much, till the liquor fill the tube, and 
 then it is lealed hermetically at the end A. Then 
 the fpirit contradls within the tube as it cools. It 
 is inclofed in a frame, which is graduated into de- 
 grees, tor heat and cold. For hot weather dilates 
 the fpirit, and makes it run further up the tube ; 
 and coid weather on the contrary, contrails it, and 
 makes it fink lower in the tube. And the parti- 
 cular divifions, fhew the feveral degrees of heat 
 and cold •, againfl the principal of which, the words 
 heat, cold, temperate, &c. are written. 
 
 They that would fee more machines defcribed, 
 may confult my large book of Mechanics, where 
 he will meet with great variety. 
 
 FINIS. 
 
 ERRATUM. 
 
 Pagt 35, Line 10 from the bottom, read. Cor. 1. Hence
 
 m: 
 
 V' c'/i^.
 
 
 Pl.K.Art?^ 
 
 r
 
 THE 
 
 PROJECTION 
 
 O F T H E 
 
 SPHERE, 
 
 Orthographic, Stereographic, 
 and Gnomoni cal. 
 
 Both demon rtrating the 
 
 PRINCIPLES, 
 
 And explaining the 
 
 PRACTICE 
 
 Of thefe three feveral Sorts of 
 Projection. 
 
 The SECOND EDITION, 
 
 Correfted and Improved. 
 
 hi Minimis Ufus- 
 
 LONDON, 
 
 Printed for J>^ Nourse, in the Strand, 
 Bookfeller in Ordinary to His Majesty* 
 
 M PCC LXIXo
 
 THE 
 
 PREFACE. 
 
 ^H E Projeclion of the Sphere, or of its Circles^ 
 has the fame relation to Spherical 'Trigonometry y 
 that pr apical Geometry has to -plane 'Trigonometry. For 
 as the one faves a deal of Calculation.^ by drawing a 
 few right Lines, fo does the other by drawing a few 
 Circles, The Projetlion of the Sphere gives a Learner 
 a good Idea of the Sphere and all its Circles, and of 
 their fever al Pofitions to one another, and confequently 
 of Spherical Triangles, and the Nature of Spherical 
 Trigonometry. 
 
 I have here delivered the Principles of three fort s^ 
 cf Projection, in afmall compafs -, and yet the Reader 
 will find here, all that is efjential to the fubjeB ; and 
 yet nothing fuperfluous •, for I think na more need he- 
 faid, or indeed can be faid about it, to make it intelli- 
 gible and pr amicable. For here is laid down, not only 
 the whole Theory, but the Practice likewife, ' Tet the 
 praulical Part is entirely difengaged from the Thecry ; 
 fo that any body {tho' he has no defire or leifure to 
 attain to the Theory,) may never thelefs, by help of the ■ 
 Problems, make himfclf Majier of the Pra^ice. For 
 which end I have endeavoured to make all the rules re- 
 lating to practice, plain, floortt and eafy, and at the 
 fame time full and clear. 
 
 It is true the foluticn of Problems this way, muf^ 
 he allowed to be imperfect ; for there will always be 
 fome errors in working., as well as in the injlruments 
 A 2.. ^^
 
 iv The P R E F A C E. 
 
 we work with. But nohody infeeking an accurate fo- 
 lution to a Problemt will trujl to a Proje^ion by ft ale 
 and compafs ; hecaufe this cannot be depended on in 
 cafes of great nicety. Tet where no great exa5lnefs is re- 
 quired it will be found very ready and ufeful -, and, 
 be/ides., willferve to prove and confirm the folution ob- 
 taind by Calculation. 
 
 But then this defeEl is abundantly recompenfed by the 
 eafinefs of this method. For by fcale and compafs only, 
 all forts of Problems belonging to the Sphere^ as in Af- 
 troncmy^ Gecgrafhy., Diallings i^c. may be folved 
 with very little trouble^ which require a great deal of 
 time and pains, to work out trigonometric ally by the ta- 
 bles. It likewife affords a great pie af lire to the mind., 
 that one can, in a little time, defer ibe the whole furni- 
 ture of Heaven, and Earth, and reprefent them to the 
 eye, in a fmall fcheme cf paper. 
 
 But its principal ufe is for fuch perfons {and that is 
 by far the greater nuuiber) as having no opportunity 
 for learning Spherical Trigonometry, have yet a defer e 
 to refolve fome Problems of the Sphere. For fuch as 
 thefe, this fraall Tree, life will be of particular fervice, 
 hecaufe the pratlical rules, efpecially of any one fort of 
 Projection, may be learned in a very little time, and 
 cire e^fiy remembered. So that I have fome hopes I 
 P^all pleafe all my Readers^ whether theoretical or prac^ 
 (ieaL 
 
 W. E- 
 
 THE
 
 t ' 1 
 
 THE 
 
 PROJECTION 
 
 OF THE 
 
 I P H E R E IN t> L A N O. 
 
 P 
 
 DEFINITIONS. 
 
 ROJECTION of the fphere is the reprd- 
 fenting its furface upon a plane, called the 
 Plane of ProjeSiion. 
 
 2. Orthographic Projedlion, is the drawing the 
 :ircles of the fphere upon the plane of fome great 
 ircle, by lines perpendicular to that plane, let fall 
 rom all the points of the circles to be proiedled. 
 
 3. The Stereographic Projection, is the drawing 
 he circles of the fphere upon the plane of one of 
 ts great circles, by lines drawn from the pole of 
 hat great circle to all the points of the circles to be 
 irojeded. 
 
 4. The Gmmonical Pi'ojecftion, is the drawing the 
 ircles of an hemifphere, upon a plane touching it 
 n the vertex, 'oy lines or rays ilTuing from the cen-' 
 ;er of the hemifphere, to all the points of the cir- 
 cles to be projefted. 
 
 5. The Primitive circle is that on whofe plane. 
 £he fphere is proje£ted. And the pole of this cir- 
 cle is called the Pole of Projeftion. The point from 
 whence thQ proje^ing righ. iines ilTue is the project- 
 i?ig Pcint. 
 
 As 6. The
 
 THE PROJECTION, &:c. 
 6. The Line of Measures of any circle is the 
 common interfeftion of the plane of projeftion, 
 and another plane that paflTes thro' the eye, and is 
 perpendicular both to the plane of projeftion, and 
 to the plane of that circle. 
 
 Scholium. 
 
 There are other Projedions of the Sphere, as the 
 Cylindrical tlie Scenogi'aphic which belongs to Per- 
 fpedive, the Glohical which belongs to Geography^ 
 Mercators, for which fee Navigation, &c. 
 
 AXIOM. 
 
 The Place of any vifible point of the Sphere upon 
 the plane of projeftion, is where the projecting 
 line cuts that plane. 
 
 Cor. If the eye he applied to the projeciing pointy 
 it ivill view all the circles of the Sphere^ and every 
 part of them^ in the projeofion^ jiifi as they appear 
 from thence in the Sphere itfelf 
 
 Scholium. 
 
 The Projection of the Sphere is only the ffiadow 
 of the circles of the Sphere upon the plane of Pro- 
 jection, the light being in the place of the eye or 
 projecting point. 
 
 The Signification of fome Characters. 
 
 -f- added to. 
 
 — fubtraCting the following quantity. 
 
 <C an angle. 
 
 zz equal to. 
 
 •J- perpendicular tOv 
 
 11 parallel to. 
 : : a proportion. 
 
 S E C T.
 
 C 3 ] 
 
 SECT. r. 
 
 72^ Orthographic ProjeEiion of the 
 Sphere, 
 
 PROP. I. 
 
 JF a right line AB is proje5led upon a plane, it is Fig.' 
 
 projected into a right line ; and its length will be tg 3, 
 the length of the projeSlion, as radius to the cojine of 
 its inclination above that plane. 
 
 For let fall the perpendiculars A^, B<^ upon the 
 plane of projeflion, then ab will be the line it is 
 projeded into ; but by trigonometry AB : is to Aa 
 or ab :: as radius : to the fine of B or cofine of ^'AB. 
 
 Cor. I. If a right line is proje5led upon a plane^ 
 parallel thereto, it is proje^ed into a right line paral- 
 lel and equal to itfelf. 
 
 Cor, 2. If an angle beprojeSied upon a plane which 
 is parallel to the two lines forming the angle ; it is 
 projected into an angle eqtial to itfe^. 
 
 Cor, 3. Any plain figure pro] e^ed upon a plane pa^ 
 rallel to itfelf, is projected into a figure fimilar and 
 equal to itfelf. 
 
 Cor. 4. Hence alfo the area of any plain figure^ is 
 to the area of its projection : : as radius, to the cofine 
 of its elevation or inclination, 
 
 PROP. II. 
 
 A circle perpendicular to the plane of proje5iio7i, is 
 projeCfed into a right line equal to its diameter. 
 
 For projecling lines drawn through all tlie points 
 of the circle fall in the common fection of the planes 
 
 A 4 of
 
 I. 
 
 4 ORTHOGRAPHIC PROJECTION 
 
 Fig. of the circle and of prqjeftion, which is a right 
 line (Geom. V. ^Ojc^nd equkl to the diameter of 
 the circle -, becaufe the planes interfedl in that dia- 
 xneten ^. E. D. 
 
 Cor. Hence any ■plane figure^ ferpendicular to the. 
 ^lane of projection is projected into a right line. For 
 the perpendiculars from every point, will all fall in 
 the cojnmon interfe5lion of the figure with the plane of 
 projection. 
 
 PROP. III. 
 
 A circle parallel to the plane of projeCIion is pro- 
 jected into, a circle eqiial to itfelf and concentric with 
 the primitive. 
 
 Let BOD be the circle, I its center, C the cen- 
 ter of the fphere, the points I, B, O, D, are pro- 
 jefted into the points C, L, F, G. And therefore 
 OICF, and BiCL are redlangled parallelograms. 
 Confequently LC — BI r: OI zi FC, (Geom, 
 III. I.). ^ E. D. 
 
 Cor. The radius CLj or CF is the coftne of the cir^ 
 cWs diflance from the primitive, for it is the fine ^/ AB. 
 
 PROP. IV. 
 
 An inclined circle is projected into an ellipjis whofe 
 tranfverfe axis is the diameter of the circle'. 
 
 Let ADBH be the -inclined circle, P its center* 
 and let it be projedLed- into adhh\ drav/ the plane 
 ABFC^ througli the center C of the fphere, per- 
 pendicular to the plane of the given circle and 
 plane of projeclion, to interfeft them in the lines 
 AB, ah; draw GPH, DE, perpendicular, and DQ 
 parallel to AB ; then becaufe the line GP, and 
 the plane of projedion are both perpendicular to 
 
 the
 
 Sea.l. OF THE SPHERE. 5 
 
 the plane ABF ; therefore GH is parallel to the Fig. 
 plane of projedion, and therefore to gh. 
 
 In the circle ADB, DQ^; =: GQH - gqh, and 
 and BP' = GP' — gp\ And (Geom. V. 12.) BP : 
 EP or DQ^: \ hp \ ep or dq^ and BP' : DQ' : : 
 hp^ : df \ that is, gf' : gqh \ \ hp^ : da[- ; and there- 
 fore agbh is an ellipfis, whofe tranfverfe gh is the 
 diameter of the circle. ^ E. D. 
 
 Cor. I. Since ah is perpendicular to gh, therefore 
 ah is the conjugate axis •, and is twice the fine of the 
 <C! AB^ to the radius gp \ that is, the conjugate axis 
 is equal to twice the cofine of the inclinanony to the 
 radius of the circle. 
 
 Cor. 2. The tranfverfe axis is equal to twice the 
 cofine of its dijtance frojn its parallel great circle. For 
 gh ii:GH 1Z2AP zz twice the fine of AK. 
 
 Cor. 3. 'The extremities of the conjugate axis are 
 dijiant from the center of the primitive, by the fines 
 of the circles neareji and greatefi dijlance from the pole 
 of the primitive. Thus aQ is the fine of AN, and 
 hQ the fine of BN. 
 
 Cor. 4. Hence alfo it is plain that the conjugate axis 
 always paffes thro" the center Q of the primitive-, and 
 is always in the line of meafures of that circle. 
 
 Scholium. 
 
 Every circle in the projeclion reprefents two equal g^ 
 circles, parallel cind equidiftant from the primitive. 
 Every right line reprefents tv/o femicircles, one to- 
 wards the eye, the other in the opnorite fide. Eve- 
 ry ellipfis reprefents two equal circles, but contra- 
 rily inclined as AB, CD ; one above the primitive 
 the other below it. 
 
 And now the Theory being laid down, it re- 
 mains only to deduce thence, fome fliort rules for 
 praftice, as follows, 
 
 PROP,
 
 Fi 
 
 o* 
 
 ORTHOGRAPHIC PROJECTION 
 
 PROP. V. Prob. 
 
 5. 21? proje^ a circle parallel to the primitive. 
 
 Rule, 
 
 Take the complement of its diftance from the 
 
 primitive, and fet it from A to E ; and with the 
 
 center C and radius CD zi perpendicular EF, de- 
 fcribe the circle D^G. 
 
 By the plain fcak. 
 
 Take the fine of its diftance from the pole of 
 the primitive ; with that radius and the center C 
 defcribe the circle. 
 
 PROP. VI. Prob. 
 
 4. To prejccl a right circle^ or one that is perpendicu- 
 lar to the plain cf projection. 
 
 Rule. 
 
 Thro' the center C of the primitive, draw the 
 diameter AB, and take the diftance from its paral- 
 lel great circle, and fet from A to E, and from B 
 to D, and draw ED, the right circle required. 
 
 By the fcale. 
 
 Take the fme of the circle's diftance from its pa- 
 rallel great circle AB, and at that diftance draw a 
 parallel ED for the circle required. 
 
 PROP. VII. Prob, 
 To proje5i a given oblique circle. 
 
 Rule. 
 
 6, Draw the line of meafures AB, and take the cir- 
 cle's neareft diftance from the primitive, and fet 
 
 from
 
 Sea.I. OF THE SPHERE, Src. 7 
 
 from B to D, upwards if it be above the primitive ; Fig. 
 or downward, if below •, likewife take its greateft 6. 
 diftance, and fet from A to E, and draw ED, and 
 let fall the perpendiculars EF, DG ; and bifed FG 
 in H, and ereft the perpendicular KHI, making 
 KH = HI = half ED ; then defcribe an ellipfis 
 ("by the Conic Sections) whofe tranfverfe is IK and 
 conjugate FG j and that fhall reprefent the circle 
 given. 
 
 By the fcale. 
 Draw the line of meafures AB ; and take the 5. 
 fmes of the circle's neareft and greateft diftance 
 from the pole of the primitive, and fet them from 
 the center C to F and G, (both ways if the circle 
 encompafs the pole, but the fame way if it lie on 
 one fide the pole \) bifecl FG in H, and ered HK, 
 HI perpendicular- to FG, and z: to the radius of 
 the circle given, or the fme of its diftance from its 
 own pole ; about the axes FG, KI defcribe an el- 
 lipfis, and it is done. 
 
 Scholium. 
 
 An ellipfis great or fmall maybe defcribed by 10. 
 points, thus ; thro' the center D of the circle and 
 ellipfis, draw ED -i- the tranfverfe AR •, land on 
 AR erefl a fufficient number of perpendiculars IK, 
 ik &c. and make as DB or DA : DE : : IK : IF : : 
 ik : if &c. then thro' all the points E, F, /, &c. 
 draw a curve. See Prop. "j^. elliplis. 
 
 PROP. VIII. Froh,. 
 To find the pole of a given ellipfis. 
 
 Rule. 
 
 Thro^ the center of the primitive C, draw the 7; 
 conjugate of the ellipfis -, on the extreme points 
 F, G, ere6t the perpendiculars FE, GD, or fet the 
 
 tranfverfe
 
 S ORTHOGRAPHIC PROJECTION 
 
 Fig. tranfverfe IK from E to D, and biled ED in R, 
 7. and let fall RP perpendicular to AB, then is F 
 the pole. 
 
 By the fcale. 
 
 7. Take CF, and CG, and apply to the fines, and 
 find the degrees anfwering or the fupplements ; 
 then take the fine of half the fum of thefe degrees, 
 if F, G be both on one fide of C, or the fine of 
 half the difference, if they lie on contrary fides % 
 and fet it from C to the pole P. 
 
 Or thus ; apply the femi-tranfverfe IH to the 
 fines, and fet the degrees from E to R i and draw 
 RP -I- to AB i and P is the pole. 
 
 PROP. IX. Proh, 
 
 'To meafure an arch of a -parallel circle-y or to fet 
 any number of degrees on it. 
 
 Rule. 
 
 With the radius of the parallel, and one foot 
 in C defcribe a circle G^, draw CGB, and Cgb j 
 then B^ will meafure the given arch G^ ; or Gg 
 will contain the given number of degrees fet from 
 B to b. So that either being given finds the other. 
 
 P R O P. X. Prob. 
 ^0 meafure any part of a right circle. 
 
 Rule. 
 
 $: In the right circle ED, let EA zz AD j and 
 let AB be to be mcafured. Make CF zz AE ; 
 widi extent BA ::;:: FG defcribe the arch GI •, draw 
 CGK to touch it in G i then is HK the meafure of 
 AB. For FG — S. <: HCK to the radius CF or 
 AE, and BA is the fame, by Cor. Prop. III. 
 
 Other*
 
 R 
 
 l,p,S.
 
 r
 
 Sea. I. OF THE SPHERE, &c. 9 
 
 Fig. 
 
 Otherwife thus. g 
 
 On the diameter ED, defcribe the femicircle 
 END, draw AN, BO, LP perpendicular to ED, 
 then ON is the meafure of BA, and NP of AL5 
 and ON or NP may be meafured as in Prop. IX, 
 
 By the fcale. 
 
 Let AL be to be meafured. Draw CD ; and 
 LM parallel to AC, then CM applied to the fines 
 gives the degrees. For radius CD : AD : : CM : 
 aL. 
 
 Cor. If the right circle pajfes thro' the center, there 
 is no more to do^ but to raife perpendiculars on ity 
 which will cut the primitive, as required. Or apply 
 the part of the right circk to the line of fines, 
 
 PROP. XI. ?roK 
 
 'To fet off any nuniher of degrees upon a right ctr^ 
 cle, DE. 
 
 Rule.^ 
 
 Draw CA -l DE, and make the < HCK = g^ 
 the degrees given, make CF zr radius AE, take 
 FG the nearefl. diitance, and fet from A to B j then 
 AB — <:; HCK, the degrees propofed. 
 
 Otherwife thus. 
 
 On ED defcribe the femicircle END, then by 
 Prop. IX. fet off NP — degrees given, draw PL 
 perpendicular to ED, then AL contains the de- 
 grees required. 
 
 Or thus hy the fcale. 
 
 Draw CD, take the given degrees of the fines, 
 and fet. from C to M, and draw ML parallel to 
 CA, then AL :;^ arch required. 
 
 PROP,
 
 10 ORTHOGRAPHIC PROJECTION* 
 
 Fig. PROP. XII. Prob. 
 
 q. To meafure an arch of an elUpfis -, or to fet any 
 lo. number of decrees upon it. 
 
 Rule, 
 About AR the tranfverfe axis of the ellipfis, 
 defcribe a circle ABR ; eredl the perpendiculars 
 BED, KFI, on AR ; then BK is the meafure of 
 EF, or EF is the reprefentation of the arch BK. 
 And BK is to be meafured, or any degrees fet up- 
 on it, as in Prop. IX. 
 
 Scholium. 
 
 Thefe Problems are all evident from the three 
 firfi: propofitions, and need no other demonflration. 
 If the fphere be projected on any plane parallel to 
 the primitive, the projeftion will be the very fame ; 
 for being effefled by parallel lines, which are al- 
 ways at the fame diitance, there will be produced 
 the fame figure, or reprefentation. Of all ortho- 
 graphic projetftions, thofe on the meridian or on 
 the folftitial colure, commonly called the Analem- 
 ma, is moft ufeful ; becaufc-a great many of the 
 circles of the fphere fall into right lines or circles, 
 whereas in the projections upon oiher planes, they 
 are projected into ellipfes, vvhich are hard to de- 
 fcribe ; which makes thefe forts of projection to be 
 negle6ted. 
 
 And by the fame rules that the circles of the 
 fpliere are pro]ed;ed upon a plane, any other figure 
 may likewife be orthographically projeded ; by let- 
 ting fall perpendiculars upon the plane from all the 
 angles, or all the points of the figure, and joining 
 thefe points with right or curve lines, as they are 
 in the figure itfelf. 
 
 By this kind of proje6tion, either the convex or 
 concave fide of the fphere, may be projedled ; 
 
 which.
 
 Sea. I. OF THE SPHERE, &c. ii 
 
 which is peculiar to this fort of projefbioft ; that Fig. 
 is, either the hemifphere towards you, or that from 9. 
 you, may be projefted upon the plane of its great 
 circle. And fince in fome cafes they both have the 
 fame appearance, it ought to be mentioned whe- 
 ther it is. 
 
 - But if both the convex and concave fides of 
 the jame hemifphere be projefted ; that is, if you 
 make two projeftions, one for the convex, the 
 other for the concave fide ; the circles in one will 
 be inverted in refped of the other, the right to 
 the left, &c. Becaufe in looking at the fame he- 
 mifphere, it will not have the fame appearance, 
 when you look at the contrary fides of it j becaufe 
 you look contrary ways at it, to fee the external 
 and internal furfaces. 
 
 SECT.
 
 [ 12 ] 
 
 Fig. 
 
 SECT. 11. 
 
 iT^e Stereographic ProjeEiion of th^ 
 Sphere. 
 
 PROP. I. 
 
 /4NT circle paffing thro* the proje5!ing pointy is 
 proje£led into a right line. 
 
 For all lines drawn from the projefting point, 
 to this circle, pafs thro* the interfedion of this cir- 
 cle and plane of projedlion, which is a right line. 
 
 Cor. I. A great circle paffmg thro* the poles of the 
 primitive is projected into a right line paffing thro* 
 the center. 
 
 Cor. 2. Any circle pajfing thro* the projeEling point 
 Is proje^ed into a right line perpendicular to the line 
 cf meafures^ and dijiant from the center^ the fend- 
 tangent of its near eft diftance from the pole oppofite to 
 12, the proje^ing point, 'thus the circle AE is projected 
 into a right line paffing thrd" G, ayid perpendicular to 
 BC, the line of meafures^ and GC is the femitangent 
 . of EM. 
 
 PROP. II. 
 
 Every circle (that paffes 7iot through the proje^ing 
 point) is proje^ed into a circle. 
 
 II. Cafe I. Let the circle EF be parallel to the 
 primitive BD -, lines drawn to all points of it from 
 the projefling point A, will form a conic furface, 
 which being cut parallel to the bafe by the plane 
 BD, the fedtion GH (into which EF is projected) 
 will be a circle by the conic fedions. 
 
 C4^.
 
 Sea. IL OF THE SPHERE. 13 
 
 Cafe II. Let BH be the line of meafures to the Fig. 
 circle EF, draw FK parallel to BD, then arch AK 12. 
 = AF, and therefore <: AFK or AHG =: AEF ; 
 therefore in the triangles AEF, AGH, the angles 
 at E and H are equal, and the angle A common ; 
 therefore the angles at F and G are equal. There- 
 fore the cone of rays AEF (whofe bafe EF is a 
 circle) is cut by fubcontrary fcftion, by the plane 
 of projedtion BD, and therefore, by the conic {tc- 
 tions, the feftion GH ("which is the projeftion of 
 the circle EF) will alfo be a circle. ^ E. D. 
 
 Cor. When AF is equal to AG, the circle EF is 
 projected into a circle equal to it felf. 
 
 For then the fimilar triangles AHG and AEF, 
 will alfo be equal, and GH zz EF. 
 
 PROP. III. 
 
 j^ny point on the fphere's fur face is projeEled into 
 a pointy dijlant from the center^ the femi-tangent of 
 its difiancefrom the pole oppofite to the projeBing point. 
 
 Thus the point E is proje6led into G, and F into 
 H ; and CG is the femi-tangent of EM, and CH 
 of MF. 
 
 Cor. I J great circle perpendicular to the primi^ 
 tive is projeSied into a line of femi-tangent s paffing thro* 
 the center^ and produced infinitely. 
 
 For MF is projected into CH its femi-tangent, 
 and EM into the femi-tangent CG. 
 
 Cor. 2. Any arch EM of a great circle per p. to the 
 primitive \ is projected into the femi-tangent of it» 
 Thus EM is projedled into GC. 
 
 Cor. 3. Any arch EMF of a great circle is pro- 
 jelled into the fum of its femi-tangent s^ of its greatefi 
 
 B ^nd
 
 ^4 stereographic projection ' 
 
 Fig. and Icaji diftances from the oppofite pole M, if it lye 
 12. on both fides of M, or the dif. of the femi-tangents^ 
 when all on one fide. 
 
 PROP. IV. 
 
 I o , The angle made by two circles on the furface of the 
 fphere is equal to that made by their reprefentatives 
 upon the plane of projeSlion. 
 
 Let the angle BPK be projected. Thro' the 
 angular point P and the center C, draw the 
 .plane of a great circle PED perpendicular to the 
 plane of projection EFG. Let a plane PHG touch 
 the fphere in P j then fince the circle EPD is per- 
 pendicular both to this plane and to the plane of 
 projection, therefore it is perpendicular to their in- 
 terfedlion GH. The angles made by circles are the 
 fame as thofe made by their tangents, therefore in 
 the plane PGH, draw the tangents PH, PF, PG 
 to the arches, PB, PD, PK -, and thefe will be 
 projeded into the lines ^H, pF, pK : Now I fay the 
 <:: HPG - <: H/)G. For the angle CPF =: a 
 right angle — Cj>A + CAP ; therefore taking a- 
 way the equal angles CPA and CAP, and cCpPF 
 ~ C^A or PpF •, confequently />F zz PF. There- 
 fore in the right angled triangles PFG and ^FG, 
 there are two fides equal and the included < right ; 
 therefore hypothenufe PG zr pG. And for the 
 fame reafon in the right angled triano;les PFH and 
 ;pFH, PH zz /)H. Laftly in the triangles PHG 
 and />HG, all the fides are refpe6tively equal, and 
 therefore < P zz </>. ^ E. D. 
 
 Cor. I . 'The rumb lines proje^ed 7nake the fame an- 
 gles with the meridians as upon the globe ; and there- 
 fore are logarithmic fpirals on the plain of the equi- 
 no6iial. For every particle of the rumb coincides with 
 fome great circle. 
 
 Cor.
 
 Sea. II. OF THE SPHERE. 15 
 
 Cor. 2. The angle made by two circles on the fphereYx^, 
 is equal to the angle made by the radii of their projec- 
 tions at the point of interfe^ion. For the angle made 
 by two circles on a plane is the fame with that made by 
 their radii drawn to the point of interfe^ion, 
 
 PROP. V. 
 
 The center of a projeEled (leffer) circle perpendicu' 
 lar to the primitive, is in the line of meafures dijlant 
 from the center of the primitive, the fecant of the lef- 
 fer circles diftancefrom its own pole ; and its radius 
 is the tangent of that dijlance. 
 
 Let A be the projeding point, EF the circle to 14; 
 be projeded, GH the projedled diameter. From 
 the centers C, D draw CF, DF, and the triangles 
 CFI, DFI are right angled at I ; then <r IFC — 
 <: FCA = 2<: FEA or 2FEG =1: 2 <:FHG - 
 <C FDG, therefore IFC + IFD - FDG + IFD 
 n a right angle ; that is CFD is a right angle, and 
 the line CD is the fecant of BF, and the radiu? 
 FD is the tangent of it. ^ E. D. 
 
 Cor. If thefe circles be aUually defcribed, *tis plain 
 the radius FD is a tangent to the primitive at F, 
 where the leffer circle cuts it* 
 
 PROP. VI. 
 
 The center of Proje^ion of a great circle is in th^ 
 line of meafures, difiant from the center of the pri^ 
 mtive, the tangent of its inclination to the primi^ 
 live ; and its radius is the fecant of its inclination. 
 
 Let A be the projedting point, EF the great cir- 15, 
 cle, GH the projeded diameter, D the center-, 
 B 2 draw
 
 i6 STEREOGRAPHIC PROJECTION 
 
 Fio". draw DA. The angle EAF being in a femicif- 
 
 15. cle is right. In the right angled triangle GAH, 
 AC is perpendicular to GH, therefore <: GAC n: 
 AHC and their double, ETB = ADC, and their 
 complements, ECF ~ CAD. Therefore CD is the 
 tangent of ECI, and radius AD its fecant. ^ E. D. 
 
 16. Cor. I. If the great oblique circle AGBH be a^u- 
 ally defcribed upon the primitive A IB. I fay, all great 
 circles pajfing thro* G will hr.ve the centers of their 
 trojetiions in the line RS drawn thro' the center D, 
 perpendicular to the line of meafures IH. 
 
 For fince all great circles cut one another at a fe- 
 micircle's diftance, all circles pafTing thro' G muft 
 cut at the oppofite point H ; and therefore their 
 centers muft be in the Line RDS. 
 
 Cor. 2. Hence alfo if any oblique circle GLH be 
 required to make any given angle with another circle 
 BGAH, // will be projeSled the fame way with re- 
 gard ^<? GAH confidered as a primitive, and RS its 
 line of Meafures \ as the circle BGxA is on the primi- 
 tive BIA, and line of meafures ID. And therefore 
 the tangent of the angle AGL to the radius GD, fet 
 from D to N, gives the center of GL. 
 
 For the <: NGD will then be equal to AGL, 
 by Cor. 2. Prop. IV. and therefore GLH is rightly 
 projefted. 
 
 Cor. 3. And for the fame reafon, if N be the cen- 
 ter of the circle G^HR ; the centers of all circles paff- 
 ing thro* g and R, will be in the line r'^s perpendi- 
 cular to \^S\ fo n is the center ofgrR, But then as 
 g, R do not reprefent oppofie points of the circle G^H, 
 therefore all circles paffing thro' g^ R, {as ^rR) will 
 ■he lejfer circles, eacept G^HR. 
 
 Scholium.
 
 11/?. i^.
 
 r
 
 Sea. ir. OF THE SPHERE. 17 
 
 Scholium. l'^g»- 
 
 Of all great circles in the prqjeftion, the primi- 
 tive is the lead. For the radius of any oblique 
 great circle (being the fecant of the inclination j is 
 greater than the radius of the primitive ; as the 
 fecant is always greater than the radius. There- 
 fore every oblique great circle in the projedion is 
 greater than the primitive. 
 
 PROP. VII. 
 
 ^he ■projeSied extremities of the diameter of any cir- 
 cle^ are in the line of meajures^ did ant frcm the cen- 
 ter of the primitive cinle^ the femi-tangents of its 
 near eft and great eft diftances from the pole of projeSiion 
 eppojite to the proje^ing point. 
 
 For the diameter of the circle EF is projeded 15. 
 into GH, from the projefling point A. But GC 17, 
 is the femi-tangent of £B, and CH the fcmi-tangent 
 of BF. ^ E. D. 
 
 Cor. I. The points where an inclined great circle i5». 
 cuts the line of meafures, within and without the pri- 
 mitioe, is dift ant from the center of the primitive^ the 
 tangent and co-tangent of half the complement of the 
 circle's inclination to the primitive. 
 
 For CG — tangent of half EB, or of half the 
 complement of IE the inclination. And (becaufe 
 the <C E A F is right) CH is the co-tangent of GAC. 
 or half EB. 
 
 Cor. 2. Hence the center D of a projected circle is jy- 
 in the line of meafures dtftant^ from the center of the i B, 
 primitive^ haf the difference of the femi-tangents of its 
 near ft and greateft diftance from the oppqfite pole, if 
 it encompaffes that pole •, but half the fum of the fe- 
 mi-tangents if it l^e on one fide the pole of proje^ion. 
 
 B 3, Cor..
 
 i8 STEREOGRAPHIC PROJECTION 
 
 Fio". Cor. 3. And the radius is half the fum of the fe^ 
 mi-tangents^ if the circle encompaffes the pole ; or half 
 the difference if it lyes on one fide. 
 
 I J. Cor. 4. Hence alfo if pq he the proje^ed poles, it 
 ivitl be qG : pG : : ^H :pH.. 
 
 For draw Gn parallel to ^A, and fince P, Q^are 
 the poles, therefore qAp is a right angle, and fince 
 the angles GAp and pKH are equal, and Gn per- 
 pendicular to A/>, therefore GA — An •, whence 
 by fimilar triangles qG : qH. : : An or AG : AH : : 
 Gp, />H, (Geom. II. 25.) And confequently the 
 line qH is cut horomonically in the points G, p. 
 
 PROP. VIII. 
 
 'the proje^ed poles of any circle are in the line of 
 meafures, within and without the primitivCy and dif- 
 ■ tant from its center the tangent and co-tangent of half 
 its inclination to the primitive. 
 
 jg^ The poles P, p of the circle EF are projected 
 into D and d ; and CD is the tangent of CAD or 
 half BCP, that is, of half GCI, the inclination of 
 the circle ICK, parallel to EF. Likewife Cd is 
 the tangent of CAi, or the co-tangent of CAD. 
 
 ^ E. D. 
 
 Cor. I . The pole of the primitive is its center ; and 
 the pole of a right circle is in the primitive. 
 
 I p. Cor. 2. The projected center of any circle is always 
 between the projected pole {nearefi to it on the fphere) 
 and the center of the primitive ; and the projected 
 centers of all circles lye between the projected poles. 
 
 For the middle point of EF or its center is pra- 
 jedled into S ; and all the points in Vp (in which are 
 all the centers) are projeded into Dd^ 
 
 Cor.
 
 0,1. 
 
 'SL.p.iS
 
 o
 
 Sea. IX. OF THE SPHERE. 19 
 
 Cor. 3. If ^ he the projected center of any circle Y'lg, 
 EFG, any right lines EG, FH faffing thro^ P will 20. 
 intercept equal arches EF, GH. 
 
 For in any circle of the fphere, any two lines, 
 pafling thro' the center, intercept equal arches ; 
 and thefe are projefted into right lines, pafTmg. 
 thro' the projected center P,^ and therefore EF,. 
 GH, reprefent equal arches. 
 
 P R O P. IX. 
 
 j^ EFGH, efgh reprefent two equal circles^ where- 20. 
 cf EFG is as far diftant from its pole P, as efg is 21.. 
 from ih& projecting point. I f^iy-, any two right lines 
 (^EP, and f¥?,) being drawn thro" v., will intercept 
 equal arches {in reprefentation) of thefe circles ; 07t 
 the fame Jiddy if V falls within the circles ; but on the 
 contrary fide, if without ;. that is, EF zi ef and 
 GH^gh. 
 
 For by the nature of the fedlion of a fphere ; 
 any two circles pafling thro'' two given points or' 
 poles on the furface of the fphere, will intercept 
 equal arches of two other circles equidiftant from 
 thefe poles. Therefore the circles EFG and ^7^ 
 on the fphere, are equally cut by the planes- of any- 
 two circles pafling thro' the proje6ling point and 
 the pole P, on the fphere. But thefe circles (by 
 Prop. I.) are proje6ted into the right lines P^ and- 
 P/, pafling thro' P. And the intercepted arches, 
 reprefenting equal arches on the fphere, are there- 
 fore equal, that is, EF — ef^ and GH zz gh. 
 
 Cor. I . If a circle is proje5ied into a right line EF, 2 2 ». 
 perpendicular to the line of meafures EG -, and if fronv 
 tJje center C a circle <?/P be defcribed pajfing thro' its- 
 pole P, and P/ be drawn ; then arch ef zz EF. And- 
 if any other circls be defcribed whofe vertex is P, the' 
 arch ef will always be equal to EF. 
 
 B 4 Gor^
 
 20 STEREOGRAPHIC PROJECTION 
 
 Fig. Cor. 2. Hence alfo^ if from the -pole of a great cir-^ 
 cle there be drawn two right lines^ the intercepted arch 
 of the projeoied great circle will he equal to the inter-' 
 cepted arch of the primitive, 
 
 20. Cor. 3. jf^fier the fame manner^ if there he two 
 24. equal circles EF, f/", whereof one is as far from the 
 pole P, as the other is from the pole of prejeBion <?, 
 cppof.te to the projcoting point. 'Then any circle drawn 
 thro' the points P, C, will intercept equal arcbY^^ 
 — ef \ and GH in ^-6, hetween it and the line of mea- 
 fures PCG. 
 
 For this is true on the fphere, and their projec- 
 tions are the fame. 
 
 Cor. 4. If from an angular point he drawn two 
 right lines thro' the poles of itsjides \ the intercepted 
 arch of the primitive^ will he equal to that angle. 
 
 For the diltance of the poles is equal to that 
 angle. 
 
 PROP. X. 
 
 jr. If QH, NK he two equal circles., whereof NK 
 2 5 is as far from the projecting point as QH. from its 
 pole P •, and if they be projected into the circles whoje 
 radii are MC or CL-, and DF cr PG, F being the 
 center of DG, and E the projected pole. I fay^ the 
 pole E will be dijiant from their centers in proportion 
 to the radii of the circles ; that is, CE : EF :. : CL : 
 DF or FG. 
 
 For fince NK and ML are parallel, and arch 
 NX zz PH, therefore <: ELI =: NKI (or ?iKl) - 
 GIP •, therefore the triangles lEL and I EG are fi- 
 milar, whence EL : EI : : EI : EG. Again the 
 angle EMI := KNI = PIQ, and therefore the 
 triangles lEM and lED are fimilar, whence EM : 
 El : : EI : ED. Therefore EP = EL X EG == 
 
 EM
 
 Sea. II. OF THE SPHERE. ai 
 
 EM X ED. Confequcntly EM : EL : : EG : ED i Fig. 
 EM + EL EM — EL 
 
 and by compofition : : : : 
 
 EG + ED EG —ED 
 
 — : •,thatis,CM :EC::FG: 
 
 2 2 
 
 FF. ^ E. D. 
 
 Cor. I. Hence if the circle KN he as far from 25. 
 the projedting point, asQYi is from either of its poles, 26, 
 and if E, O, be its proje5led poles ; then witl EL : 
 EM : : ED : : EG : : OD : OG. 
 
 This follows from the foregoing demonftration, 
 and Cor. 4. Prop. VII. 
 
 Cor. 2. Hence alfo if F he the center, and FD the 25. 
 radius of any circle QH, and E, O the projected 16. 
 poles i then EF : DF : : DF : FO. 
 
 FG + ED 
 
 For it follows from Cor. i, that — — — ; 
 
 E£j:iI5 : : OG + OD : 2^^l£'2/ 
 
 2 2 
 
 Cor. 3. Hence if the circle DBG, he as far from 27. 
 
 its proje^ed pole P, as LMN is from the projeEling 28. 
 point ; and if any right lines be drawn thro' P, as 
 MPG, NPK, they will cut off fimilar arches GK, 
 MN in the two circles. 
 
 For from the centers C, F, draw the lines CN, 
 FK, then fince the angles CPN, and FPK are 
 equal, and by this Prop. CP : CN : : FP : FK ; 
 therefore (Geom. II. 16. J ; the triangles PCN and 
 PFK are fimilar -, and the angle PCN z= < PFK; 
 therefore the arches MN and GK are fimilar. 
 
 Cor. 4. Hence alfo if thro" the projeBed pole ^ of 2j, 
 any circle DBG, a right line BPK be drawn. Then 28. 
 I fay the degrees in the arch GK fjall be the meafure 
 of DB in the proje^licn. And the degrees in DB, 
 fhall be the meafure of GK in thi projeilion. 
 
 For
 
 .*2 STEREOGRAPHIC PROJECTION 
 
 Fig. For (by Prop. IX.) the arch MN is the meafure 
 of DB, and therefore GK which is fimilar to MN, 
 will alfo be the meafure of it. 
 
 Cor. 5. ^he centers of all proje5led circles are all 
 beyond the proje^ed poles (in refpe^ to the center of 
 the primitives) ; and none of their centers can fall 
 between them, 
 
 20. Cor. 6. Hence it follows (by Cor. 5. and Pr. VIII. 
 Cor. 3.J that all circles that are not parallel to the 
 primitive have equal arches on the fphere reprefented 
 by unequal arches on the plane of proje5lion. 
 
 For if P be the proje(^ed center, then GH is 
 
 greater than EF. 
 
 Scholium. 
 
 It will be eafy by the foregoing proportions ta 
 defcribe the reprefentation of any circle, and the 
 reverfe will eafiiy fhow what circle of the fphere any 
 projedled circle reprefents. What follows here- 
 after is deduced from the foregoing propofitions, 
 and will eafily be underftood without any other 
 demonftration. 
 
 If the fphere was to be proje(5led on any plane 
 parallel to the primitive, 'tis all the fame thing. 
 For the cones of rays ilTuing from the projecting 
 point, are all cut by parallel planes into fmiilar fec- 
 tions, it only makes the projeftions bigger or lefs> 
 according to the diftance of the plane of projeftion, 
 whilft they are flill fimilar •, and amounts to no more 
 than proje<5ling from different fcales upon the fame 
 plane. And therefore the projecting the fphere on 
 the plane of a leffer circle is only projecting it upon 
 the great circle parallel thereto, and continuing all 
 the lines of the fcheme to that leffer circle. 
 
 PROP,
 
 30. 
 
 7,1%.
 
 r .
 
 Seft. II. OF THE SPHERE. ^1 
 
 Fig. 
 PROP. XL Proh, 
 
 'to draw a circle parallel io the primitive at a given 
 dijtancefrom its pole. 
 
 Rule. 
 
 Thro' the center C draw two diameters AB, DE, 29,' 
 perpendicular to one another. Take in your com- 
 pafies the diftance of the circle from the pole of the 
 primitive oppofite to the projeding point, and fee 
 It from D to F ; from E draw EF to interfed AB 
 in I i with the radius CI, and center C, defcribe the 
 circle GI required. 
 
 By the plain Scale, 
 
 With the radius CI, equal to the feml-tangent of 
 the circles diftance from the pole of projedtion op- 
 pofite the projefting point, defcribe the circle IG. 
 Here the radius of projediion CA, is the tangenc 
 of 45°» or the fem-itangent of 90°, 
 
 PROP. XII. Proh. 
 
 TV draw a lejfer circle perpendicular to the primitive 
 fit a given dijtance from the pole of that circle. 
 
 Rule. 
 
 Thro' the pole B draw the line of meafures AB, 3^* 
 make BG the circle's diftance from its pole, and 
 draw CG, and GF perpendicular to it ; with the ra- 
 dius FG defcribe the circle GI required. 
 
 By the Scale. 
 
 Set the fecant of the circle's diftance from its pole 
 from C to F, gives the center. With the tangent 
 of that diftance for a radius, defcribe the circle GI. 
 
 Or thus, make BG the circle's diftance from its 
 pole J and GF its tangent, fet from G, gives F the 
 
 center j
 
 24 STEREOGRAPHIC PROJECTION" 
 
 Fig. center •, thro' G defcribe the circle GI from the cen« 
 30. ter F. 
 
 Cor. Hence a great circle perpendicular to the pri' 
 mi live, is a right line CDE drawn thro' the center 
 'perpendicular to the line of meafures. 
 
 Scholium. 
 
 When the center F lyes at too a great a diftance ; 
 draw EG, to cut AB in H i or lay the femi-tan- 
 gent of DG from C to H. And thro' the three 
 points G, H, I, draw a circle with a bow. 
 
 PROP. XIII. Prob. 
 
 ^0 defcribe an oblique circle at a given difiance from 
 a pole given. 
 
 Rule, 
 
 21. Draw the line of meafures AB thro' the given 
 point />, if that point is given ; and draw DE J- 
 to it, alio draw E/)P. Or if the point p is not gi- 
 ven, fet the height of the pole above the primitive 
 from B to P. Then from P let of PH =: Pi z= cir- 
 cle's diftance from its pole ; and draw EH, EI, to 
 interfecl A B in F and G. About the diameter FG 
 dcicribe the circle required. 
 
 By the Scale. 
 
 If the point P is given, apply Cp to the femi-tan- 
 gents and it gives the diftance of the pole from D, the 
 pole of projeclion oppofite to the projefting point. 
 This diftance being had, you'll eafily find the great- 
 eft and neareft diftances of the circle from the pole 
 of the primitive oppofite to the proje6ting point ; 
 take the femi tangents of thelc diftances and fetfrom 
 C to G and F, both the fame way if the circle lye 
 all on one fide, but each its own way, if on different 
 fides of D. And then FG is the diameter of the 
 circle required to be drawn. 
 
 Cor*
 
 Sea. II. OF THE SPHERE. 25 
 
 Cor. I. If ¥ be the pole of a great circle as o/Fig. 
 
 DLE. Draw EFH, and make HP — DH, and 31. 
 
 draw E^P, and then P is its center. 
 
 Or thus ^ draw EFH thro' the pole F, make HK 
 
 00 degrees •, draw EK cutting the line of msafures in 
 
 L. ^hrd' the three points D, L, E, draw the great 
 
 circle required. 
 
 Cor. 2. Hence it willbeeafy to draw one circle pa^ 
 rallel to another* 
 
 PROP. XIV. Proh. 
 ^hro^ two given points A, B, to draw a great circle. 
 
 Rule. 
 
 Thro' one of the points A, draw a line thro' the 32. 
 center, ACG ; and EF perpendicular to it. Then 
 draw AE, and EG perpendicular to it. Thro' the 
 three points A, B, G draw the circle required. 
 
 Or thus; From E (found as before) draw EH, 
 and then HCI, and laftly EIG, gives G a third 
 point, thro' which the circle muft pafs. 
 
 By the Scale. 
 
 Draw ACG ; and apply AC to the femi-tangents, 
 find the degrees, fet the femi-tangent of its fupple- 
 ment from C to G, for a third point. 
 
 Or thus % Apply AC to the tangents, and fet the 
 tangent of its complement from C to G. And 
 thro' the three points ABG, defcribe the circle re- 
 quired. ' 
 
 For fince HEI or AEG is a right angle, there- 
 fore A, G are oppofite points of the fphere j and 
 therefore all circles pafTing thro' A and G arc 
 great circles. 
 
 Scholium. 
 
 If the points A, B, G lie nearly in a right line, 
 then you may draw a circle thro' them with a bow. 
 
 PROP.
 
 26 STEREOGRAPHIC PROJECTION 
 
 Fig. 
 
 P R O P. XV. Prch. 
 
 About a pole given, to defcrihe a circle thro^ a given 
 point. 
 
 Rule. 
 
 23. Let P be the pole, and B the given point ; thro* 
 P, B defcribe the great circle AD (by Prop. XIV.), 
 whofe center is E ; thro' the center C draw CPH j 
 and from the center E, draw EB, and BF perpen- 
 dicular to it. To the center F, and radius FB 
 defcribe the circle BGH required. 
 
 PROP. XVI. Prok 
 
 'to find the poles of any circle FNG. 
 Rule. 
 r.1^ Thro' its center draw the line of meafures AG, 
 and DE perpendicular to it. Draw EFH, and 
 fet its diftance (from its own pole) from H to P, 
 and draw E/)P, then /> is the pole. 
 
 Or thus. Draw EFH, EIG, and bifefl HI in 
 P, and draw E^P, and p is the internal pole. Laft- 
 ly draw PCQ, and EQ^, and q is the external pole. 
 
 In a great circle DLE, draw ELK, and make 
 DH - AK, (or KH - AD, and draw EFH, and 
 F is the pole. 
 
 By the Scale. 
 
 Apply CF to the femi-tangents, and note the 
 dep-rees. Take the fum of thefe degrees and of 
 the circle's diftance from its pole, if the circle lie 
 all on one fide, but their difference if itencom- 
 pafles the pole of projeftion -, fet the femi-tangent 
 of this fum or difference from C to the internal 
 pole p. And the femi-tangent of its fupplement 
 Cq, gives the external pole q. 
 
 Or thus^ Apply CF and CG to the femi-tangents, 
 fet the femi-tangent of half the fum of the degrees
 
 JD 
 
 V. p.%^.
 
 c
 
 Se-a. I!. OF THE SPHERE. ij 
 
 {\i the circle lies all one way) or of half the dif- Fig. 
 ference (if it encompalTes the pole of projedion), 31. 
 from C to the pole p j and the femi-tangent of the 
 fupplement, C^ gives the external pole q. 
 
 In a great circle as DLE, draw the line of mea- 
 fures AB perp. to DE -, and let the tangent and 
 co-tangent of half its inclination, from the center 
 C, different ways to F and /; which gives the in- 
 ternal and external poles F and /. 
 
 PROP. XVII. Proh, 
 
 *To draw a great circle at any given inclination above 
 the primitive ; or making any given angle with it^ at 
 a given point. 
 
 Rule. 
 
 Draw the line of meafures AB j and DCE per- 34^ 
 pendicular to it. Make EK — 2HD — twice the 
 complement of the circle's inclination ; (ov DK — 
 2 AH — twice the inclination j ; and draw EKF, 
 then F is the center of EGD, the circle required. 
 
 Or thus •, Draw D£ and AB perp. to it, and let 
 D be the point given. 'Make AH the inclination, 
 and draw EGH and HCN ; and END, to cut AB 
 in O. Then bifecl GO in F, for the center of the 
 circle required. 
 
 By the Scale. 
 
 Set the tangent of the inclination in the line of 
 meafures from C to F, then F is the center. Stt 
 the femi-tangent of the complement from C to G j 
 then GF or DF is the radius. 
 
 Or the fecant of the inclination let from G or D 
 •to F gives the center. 
 
 Cor. To draw an oblique circle to make a given an- 
 gle with a given oblique circle DGE at D. Draw 
 EGH, and fet the given angle from H to I, and draw 
 ELI. "thro' D, Lj E defcrihe a great circle. 
 
 PROP.
 
 28 STEREOGRAPHIC PROJECTION 
 
 Fig. 
 
 PROP. XVIII. Proh, 
 
 through a given point P, to draw a great circle^ 
 to make a given angle with the -primitive. 
 
 Rule. 
 
 25' Thro* the point given P and the center C draw 
 the line AB •, and DE perpendicular to it. Set the 
 given angle from A to H and from H to K, and 
 draw BGK j with radius CG, and center C defcribe 
 the circle GIF; and with radius BG and center P 
 crofs that circle in F. Then with radius FP and 
 center F, defcribe the circle LPM required. 
 
 By the Scale, 
 
 With the tangent of the given angle and one 
 foot in Q defcribe the arch FG. With the fecant 
 of the given angle and one foot in the given point 
 P, crofs that arch at F. From the center F def- 
 cribe a circle thro' the point P. 
 
 PROP. XIX. Proh. 
 
 To draw a great circle to make a given angle with 
 a given oblique circle FPR, at a given point P, z« 
 that circle. 
 
 Rule. 
 
 56. . Thro' the center C and the given point P, draw 
 the right line DE •, and AB perpendicular to it; 
 draw AfG and make BM = 2DG -, and draw AM 
 to cut DE in 1. Draw IQ^ perpendicular to DE, 
 then IQ^is the line wherein the centers of all cir- 
 cles are tound which pafs thro' the point P. Find 
 N the center of the given circle FPR, and make 
 the angle NPL equal to the given angle, then L 
 is tlie center of the circle HPK required. 
 
 By
 
 P-A\
 
 /^^
 
 Sea. II. OF THE SPHERE. 2^ 
 
 By the Scale, Fig. 
 
 Thro' P and C draw DE ; apply CP to the femi- 3^* 
 tangents, and let the tangent of its complement 
 from C to I (or the fecant from P to I) On DI 
 ered the perpendicular IQ^ Find the center N of 
 FPR, and make the angle NPL zz angle given, 
 and L is the center. 
 
 Cor, If one circle is to he drawn perpendicular to ■ 
 another^ it mufi be drawn thro* its poles, 
 
 PROP. XX. Proh. 
 To draw a great circle thro' a given point P, to make 
 a given angle with a given great circle DE. 
 
 Rule, 
 About the given point P as a pole (by Prop. 13. Z7,. 
 Cor. I.) defcribe the great circle FG ; find I the pole 
 of the given circle DE, and (by Prop. 16.) about 
 the pole' I (by Prop. 13.) defcribe the fmall circle 
 HKL at a diftance equal to the given angle, to in- 
 terfed FG in H •, about the pole H defcribe (by 
 Prop. 13.) the great circle APB required. 
 
 PROP. XXI. Proh. 
 To draw a great circle to cut two given great circles 
 chd, ebf at given angles. 
 
 Rule, 
 Find the poles s, r, of the two given circles, 50; 
 by Prop. 16. about which draw two parallels phk, 
 fnk, at the diftances refpeftively equal to the an- 
 gles f^iven by Prop. 13. the point of interfed ion P, 
 is the pole of the circle w<?^ required^ 
 
 Cor. Hence^ to draw a right circle to^ make with 
 an oblique cirde, abd, any given angle. Draw 
 a parallel phk at a diftance from the pole of the ob- 
 limie circle^ equal to the given angle. Its interJe5iion 
 
 c ■ /«^//^
 
 30 STEREOGRAPHIC PROJECTION 
 
 Yig.f with the primitive^ gives the pole of the right circle 
 get required. 
 
 PROP. XXII. Prok 
 
 'To lay any number of degrees on a great circle, or 
 to meafure any arch of it. 
 
 Rule, 
 
 og Let AFI be the primitive ; find the internal pole 
 ' P of the given circle DEH (by Prop i6.) lay the 
 degrees on the primitive from A to F, and draw 
 PA, PF, intercepting the part required DE. Or 
 to meafure DE, draw PEF and PDA, and AF is 
 its meafure, and applied to the line of chords (hows 
 jiow many degrees it is. 
 
 Or thus i Find the external pole p of the given 
 circle, fet the given degrees from 1 to K, and draw 
 pi, pK, intercepting the part DE required. Or to 
 meafure DE, thro' D and E draw />!, pK, then Kf 
 is the meafure of DE. 
 
 Or thus ; Thro' the internal pole P, draw the 
 lines DPG, and EPL -, fetting the given degrees 
 from G to L in tiie circle GL \ then DE is the arch 
 required. Or if DE be to be meafured, then the 
 degrees in the arch GL is the meafure of DE. 
 
 Or thus •, Set the given degrees from G to H in 
 the circle GL and from the external pole P, draw 
 pG, pH, intercepting DE the arch required Or 
 to meafure DE, draw pDG, />EH, then the degrees 
 in GH, is equal to DE. 
 
 By the Scale for right Circles. 
 
 3^' Let CA be the right circle, take the number of 
 degrees off the femi-tangents and fet from C to D 
 for the arch CD. Or if the given degrees are to 
 1)€ fet from A, then take the degrees off the fcmi- 
 "tan<2;ents from qc° towards the be2;inning, and fet 
 froni A to D. And if CD was to be meafured, 
 
 apply
 
 Sea. 11. OF THE SPHERE. 31 
 
 apply it to the beginning of the femi-tangents •, and Fig» 
 to meafure AD, apply it from 90° backwards, and 
 the degrees intercepted gives its meafure. 
 Scholium. 
 The primitive is meafured by the line of chords, 
 or elfe it is actually divided into degrees. 
 
 PROP. XXIII. Proh. 
 
 ^0 fet any number of degrees en a lejjfer circle^ or 
 to meafure any arch of it. 
 
 Rule. 
 Let the lefTer circle be DEH ; find its internal pole 38." 
 P, by Prop. 1 6. defcribe the circle AFK parallel to 
 the primitive, by Prop. 1 1. and as far from the pro- 
 jecting point, as the given circle DE is from its 
 internal pole P, fet the given degrees from A to - 
 F, and draw PA, PF interfering the given circle 
 in D, E •, then DE is the arch required. Or to 
 meafure DE, draw PDA, PEF, and AF Ihows 
 the degrees in DE. 
 
 Or thus ; Find the external pole ^, of the given 
 circle by Prop. 16. defcribe the lelTer circle AFK 
 as far from the projecting point, as DE the given 
 circle is from its pole p, by Prop. 11. fet the de- 
 grees from I to K and draw pD\, ^EK, then DE 
 reprefents the given number of degrees. Or to 
 meafure DE ; draw /^DI, ^EK ; and KI is the mea- 
 fure of DE. 
 
 Or thus i Let O be the center of the given circle 
 DEH-, thro' the internal pole P, draw lines DEG, 
 EPL, divide the quadrant GQ^into 90 equal de- 
 grees, and if the given degrees be (0.1 from G to L, 
 then DE will reprefent thefe degrees. Or the de- 
 grees in GL will meafure DE. 
 
 Or thus ; Divide the quadrant GR into 90 equal 
 
 parts or degrees, and fet the given degrees from G 
 
 to H, and draw pDG^ />EH, from the external pojie 
 
 p i then DE will reprefent the given degrees. Or 
 
 C 2 ' thro'
 
 3^ STEREOGRAPHIC, &c. 
 
 Fig. thro' D, E drawing />DG, />EH, then the number 
 of equal degrees in GH is the meafure of DE. 
 Scholium. 
 Any circle parallel to the primitive is divided or 
 meafured, by drawing lines from the center, to the 
 like divifions of the primitive. Or by help of the- 
 chords on the fedtor, fet to the radius ot' that circle. 
 
 PROP. XXIV. Prob, 
 I'd meafure any angle, ' 
 Rule. . 
 
 By Cor. i. Prop. 13. About the angular point as 
 a pole, defcribe a great circle, and, note where it 
 interfefts the legs of the angle ; thro' thefe points of 
 interfeftion, and the angular point, draw two right 
 lines, to cut the primitive ; the arch of the primi- 
 tive intercepted between them is the meafure of the; 
 angle. This needs no example. ■ 
 
 Or thus\ by Pi'op. 16. Find the two poles of 
 the containing fides, ('the neareft, if it be an acute 
 angle, otherwife the furtheit) and thro' the angular 
 point and thefe poles, draw right lines to the primi- 
 tive, then the intercepted arch of the primitive is 
 31. the angle required. As if the angle AEL was re- 
 quired. Let C and F be the poles of EA and EL. 
 From the aiigular point E, draw ECD and EFH. 
 Then the arch of the primitive DH, is the meafure 
 ofthe angle AEL. ^^ /^ '^■^- • 
 
 Scholium. 
 
 Becaufe in the Stereographic Projection of the 
 Sphere, all circles are projefted either into circles or 
 right lines, which are eafily defcribed •, therefore 
 this fort of proje6lion is preferred before all others. 
 Alfo thofe planes are preferred before others to pro- 
 ject upon, where moil circles are projeftcd into 
 .right lines, they being cafier to defcribe and meafure 
 Jihan circles are-, fuch are the projections on' the 
 •planes of the meridian and folftitial colure. 
 
 'SECT.
 
 £23 I 
 
 SECT. IIL ^"^^ 
 
 Tie Gnomonical Pf^ojeSiion (j/*/^^ Sphere. 
 
 P R O P. L 
 
 Every great circle ^j BAD is proje^ed into aright 39* 
 lijie, perpendicular to the line of meafures^ and dif- 
 tant from the center^ the co-tangent of its inclination^ 
 or the tangent of its near eft diftance from the pole of 
 proje£iion. ? 
 
 Let CBED be perpendicular both to the givem 
 circle BAD and plane of proje6lion,' and then the 
 interfediion CF will be the line of meafures. Now 
 fince the plane of the circle BD, and the plane of 
 projedlion are both perpendicular to BCDE, ther£»- 
 fore their common feftion will alfo be perpendicu- 
 lar to BCDE, and confequently to the line of mea- 
 fures CF. Now fmce the projefting point A is in 
 the plane of the circle, all the points of it will be 
 proj':£led into that feAion ; that is, into, a right 
 line paffing thro' d^ and perpendicular to ( d. And 
 Cd is the tangent of QDy or co-tangent of CdA, 
 ^ E. D. 
 
 Cor. I. A great circle pe7'pe7Tdictuar to the plane of ^^i 
 prcjeSlicn is projetied into a right line paffing thro'' 
 the cente of projection \ and any arch is projected in- 
 to its correfpondent tangent. 
 
 Thus the arch CD is projefled into the tangent C<5?. 
 
 Cor. 2. Any point as D, cr the pole of any circky 
 is projected into a point d difiant from the pole of p^a- 
 jeotio.i C, ihe tangent of that difiance. 
 
 Cor» 3. If tisjo great circles he perpendicular, to each 
 
 other, and cm of them paffes thvo' the pole of project 
 
 C 3 tiort'^.
 
 34 GNOMONICAL PROJECTION 
 
 Fig. tion j they will be proje£fed into two right lines per- 
 39. pendunlar to each other. 
 
 For the reprelentation of that circle which pafles 
 thro' the pole of projedion is the line of meafures 
 of the other circle. 
 
 Cor. 4. And hence if a great circle he perpendicu- 
 lar to fever al other great circles^ and its reprefentation 
 pafs thro' the center of -projedion ; then all thefe cir- 
 cles will he reprefented by lines parallel to one another^ 
 and perpendicular to the line of meafures or reprefen^ 
 iation of that frfi circle. 
 
 P R O P. II. 
 
 2Q« If i'^o great circles interfe5l in the pole of projec- 
 tion \ their reprefentations fhall make an angle at the 
 center of the plane of projection equal to the angle 
 made by thefe circles on the fphere. 
 
 For flnce both thefe circles are perpendicular to 
 the plane of projeftion -, the angle made by their 
 interfedions with this plane, is the fame as the angle 
 made by thefe circles. ^ E. D, 
 
 PROP. III. 
 
 Any leffer circle parallel to the plane of projeSfion is 
 proje5led into a circle^ whofe center is the pole of pro- 
 jection ; and radius the tangent of the circlets dijlance 
 from the pole of projection. 
 
 39' 
 
 Let the circle PI be parallel to the plane GF, 
 then the equal arches PC, CI are projeded into the 
 equal tangents GC, CH •, and therefore C the 
 point of contad and pole of the circle PI and of 
 the projedion, is the center of the reprefentation 
 GH. 4 ^- D. 
 
 Cor. Jf a circle be parallel to the plane of projec- 
 iionj and 45 degress from the pole^ it is projected into. 
 
 ^ (irch
 
 
 
 
 
 
 
 38. 
 
 K 
 
 ■tr 
 
 
 y<rT 
 
 \ 
 
 OMI 
 
 ^ 
 
 /'^ 
 
 A dI 
 
 CP 
 
 ^ 
 
 / 1 
 
 
 ^"^C 
 
 
 jyL 
 
 
 
 
 Q 
 
 
 ^l./;.34.
 
 SIICAL PROJECTION 
 
 be projtliid into two right lines pir- 
 
 > other. 
 
 eniation of that circle which paSes 
 
 f projedion is the line of meafures 
 
 lie. 
 
 hettce if a great circle he perpendicu- 
 ler great circles, and its reprejinlation 
 Iter of projeiiion ; then all tbefe cir~ 
 fmted by lines parallel to one another ^ 
 r to the line of meafures or reprefea- 
 ji circle. 
 
 ■*-Tl O P. II. 
 Jrcki inlerfeU in the pole of projec- 
 tfmtatiovs jhall make an angle at tht 
 Esflj of proje^ion equal to the angle 
 •du on the fpbere. 
 
 h thelc circles are perpendicular to 
 ojeftion ; the angle made by their 
 h this plane, is the fame as the angle 
 ircles. j^ £. £». 
 
 PROP. in. 
 
 ? parallel to the plane of preje^ion is 
 rc/e, whofe center is the pole of pro- 
 us the tangent of the circle's dijlance 
 ^rojeStion. 
 
 e PI be parallel to the plane GF, 
 ches PC, CI arc projefted into the 
 GC, CH ; and thiTefore C the 
 and pole of the circle PI and of 
 !S the center of the repreicntation 
 
 ff be parallel to the plane of projee- 
 
 eis from tht pole, it is projeSed into 
 
 » lirth 
 
 "^l/P-;i4. 
 
 o
 
 Sea. Iir. OF THE SPFIERE. 35^ 
 
 a circle equal to a great circle of the fphere •, and may Fig^ 
 therefore be looked upon as the primitive circle in this 
 projection, and its radius the radius of projeClion, 
 
 PROP. IV. 
 
 Every leffer circle {not parallel to the plane of pro- 4.0. 
 jetlion) is proje£led into a conic feCiion., ivhofe tranf- 
 'verfe axis is in the line of meafures^ and whofe nearefi 
 vertex is dijiant from the center of the plane the tan- 
 gent of its near eft difiance from the pole of projeUion ; 
 and the other vertex is dijiant the tangent of its furthefl: 
 dijtance. 
 
 Let BE be parallel to the line of meafures dpy 
 then any circle is the bafe of a cone whofe vertex, 
 is at A, and therefore that cone being produced 
 will be cut by the plane of projeilion in fome conic 
 fection ; thus the circle whofe diameter is DF will 
 be cut by the plane in an ellipfis whofe tranfverfe is 
 df\ and Cd is the tangent of CAD, andC/of CF. 
 In like manner the cone AFE being cut by the 
 plane, / will be the neareft. vertex \ and the other 
 point into v/hich E is projedled is at an infinite 
 d. fiance. Alfo the cone AFG (whofe bafe is the 
 circle FG) being cut by the plane / is the neareft ; 
 vertex i and GA being produced gives d the other 
 vertex. ^ E. D. 
 
 Cor. I. If the dijlance of the furthefi point of the 
 circle be lefs than 90^ from, the pole of projeSfion^ . 
 then it will be projeEled into an ellipfis. 
 
 Thus DF is projefted into ///^ and DC being lefs . 
 than 90°, the feftion df is an ellipiis, whofe ver- 
 tices are at d and /•, for the plane df cuts both fides 
 of the cone, iA, /A. 
 
 Cor. 2. // the furthefl point be more-than-^o de- 
 grees from the pole of projcution^ it will be prcje^ed 
 C 4. intm
 
 36 GNOMONICAL PROJECTION 
 
 Fig. into an hyperbola. Thus the circle FG is projeoied ints 
 
 40. an hyperbola whofe vertices are f and d, and tran- 
 verfe fd. 
 
 For the plane dp cuts only the fide A/ of the cone. 
 
 Cor. 3. ji^nd in the circle EF, where the furtheji 
 point E is 90'' from C -, it will be projected into a pa- 
 rabola^ whofe vertex is f. 
 
 For the plane dp (cuttmg the cone FAE) is pa- 
 rallel to the fide AE. 
 
 Cor. 4. If H be the center^ andK, k^ I, the fo- 
 cus of the ellipfis,' hyperbola^ or parabola ; then HK rz 
 
 — ■' for the elipftSy and Hk — ."i; — L 
 
 2 2 
 
 for the hyperbola ; and (drawing fn perpendicular on 
 
 AEJ // zz — — L , for the parabola •, which are 
 
 2 
 
 the reprefentations of the circles DF, FG, FE re- 
 
 fpe^iively. 
 
 This all appears from the Conic Sections. 
 
 PROP. V. 
 
 41. Let the plane TW he perpendicular to the plane of 
 projeBicn TV, and BCD a great circle of the Jphere 
 in the plane TW. And let the great circle BED be 
 projeBed in the right line bek. Draw CQS -^ hk, 
 and Cm \\ to it and equal to CA, and make QS iz Q;;^ j 
 ihen I fay any angle QJt — Q£. 
 
 Suppofe the hypothenufe AQ^to be drawn, then 
 fince the plane ACQ^is perpendicular to the plane 
 Ti', and ^^Q is -^ to the interfe£lion CQ, therefore 
 ^Q^is perpendicular to the plane ACQ^ and confe- 
 quently i'Q^is perpendicular to the hypothenufe 
 AQ^ But AQ z= Qw — Qj, and Q5 is alfo perpen- 
 dicular to bCl, Therefore all angles made at S cut 
 the line bCl in the fame points as the angles made 
 
 at
 
 Sea. III. OF THE SPHERE. 37 
 
 at A ; but by the angles at A the circle BED is Fig. 
 projeded into the line 3Q. Therefore the angles 41. 
 at s are the meafures of the parts of the projefted 
 circle ^Q ; and s is the dividing center thereof. 
 
 Cor. I . y^;?y gj'Cdt circle /QZ" is projeSied into a line 
 cf tangents to the 'radius SQ^. 
 
 For Q£ is the tangent of the angle QS/ to the 
 radius QS or Qnt. 
 
 Cor. 2. If the circle hC paf: thro'' the center of 
 projection ; then A the projecling point is the divid- 
 ing center thereof. And Qb is the tangent of its cor- 
 refpondent arch C3, to CK the radius of projeciion. 
 
 PROP. VI. 
 
 'Let the parallel circle GEH he as far from the 41. 
 pole of projection C as the circle FKI is from its pole 
 P i and let the difiance of the poles C, P be hifethd 
 by the radius AO, and drazv bAD perpendicular to 
 AO •, then any right line bek drawn thro' />, will cut' 
 off the arches hi zz. Fa/, a?;d ge zz kf (fuppoftng f the 
 other vertex)., in the reprefen:ations of thefe equal cir- 
 cles in the plane of projetiion. 
 
 _ For let G, E, R, L, H, N, R, K, I be refpec- 
 tively projefted into the points g^ e, r, /, h, w, r, 
 ^, /. Then fince in the fphere,- the arch BF zz 
 DH, and arch BG zr DI. And the great circle 
 BEKD makes the angles at B and D equal, and is 
 projeded into a right line as bl ; therefore the tri- 
 an^.ular figures BFN and DHL are fimilar, and 
 equal ; and likewife BGE, and DIK are fimilar and 
 equal, and LH zz NF, and KI =: EG ; whence 
 it is evident their projedlions Ih — nFj and kf = ge. 
 ^E.D. 
 
 PROP.
 
 35 GNOMONICAL PROJECTION 
 
 PROP. VII. 
 
 Fig. 
 
 42. ^f ^h ^^^ ^^^^ ^^ ^^^ projections of two equal cir^ 
 cles, whereof one is as far from its pole P as the other 
 from its pole C j which is the center of proje5fion ; and 
 if the dijlance of the proje£ied poles C, p be divided 
 in 0, fo that the degrees in C^, op^ be equal, and the 
 perpendicular oS be ere Bed to the line of meafures gh, 
 I fay the lines pn, CI, drawn from the poles C, p 
 thro* any pomt QJn the line <?S, will cut off the arch 
 F« = <. QCp. 
 
 For drawing the great circle GPI, in a plane per- 
 pendicular to the plane of projection. The great 
 circle AG perpendicular to CP is projected into 
 oS by Prop. 1. Cor. 3. Now let Q^ be the pro- 
 jeftion of q, and fince p(\, CQ are right lines, 
 therefore they reprefent the great circles P^, Cq. 
 But the fpherical triangle P^C is an ifoceles-trian- 
 gle, and therefore the angles at P and C are equal. 
 But becaufe P is the pole of FI, therefore the 
 great circle P^ continued, will cut an arch otf FI 
 = <: CP^ = <: PC^ = <: QCp by Prop. II. 
 That is Cfmce ¥n represents the part cut off from 
 FI) arch F« - arch Ih or <: QQh. ^ E, D. 
 
 Cor. Hence if from the proje5!ed pole p of any cir- 
 cle, a perpendicular be erected to the line of meafures ; 
 it will cut off a quadrant from the reprefent ation of 
 that circle. 
 
 For that perpendicular will be parallel to OS ; 
 Q^being at an infinite diftance. 
 
 PROP,
 
 Seft. HI. OF THE SPHERE. 39 
 
 Fig. 
 PROP. VIII. 
 
 Let IFnk be the -projeBion of any circle FI, and p 42, 
 the projected pole P. And if Cg be the co- tangent of 
 CAt*, nnd gB perpendicular to the line of meafures 
 ^C, and CAP be bifcoted by AO, and the line 0^^ be 
 draijun to any point B, and alfo pB cutting Fnk in d, 
 J fay the angle goB :zz arch Fd. 
 
 For the arch PG is a quadrant, and the < go A. 
 
 — <gpA -f- < oAp — (becaufe GCA andgAp arc 
 right angles) ^AC -\- oAp zz. gAC -\- CAo zz <; 
 gAo. Therefore gA zzgo, confequently is the 
 dividing center ofgB the reprefentation of GA -, and 
 confequently by Prop. V. <!,goB is the meafure of 
 ^B. But fince pq reprefents a quadrant, therefore 
 p is the pole of ^'■B, and therefore the great circle 
 pdQ pafTing thro' the pole of the circles gB and F^a 
 will cut off equal arches in both, that is Fd —gB 
 
 goB. Gy E. D. 
 
 Cot. The <C goB is the meafure of the angle gpB. 
 
 For the triangle gpB reprefents a triangle on the 
 fphere wherein the arch which ^B reprefents is equal 
 to the angle which <Z p reprefents, becaufe gp is 
 90 degrees. Therefore goB is the meafure of both, 
 
 Scholium. 
 
 Thus far I have treated of the theory; what 
 follows is the praftical part, and depends altoge- 
 ther on what is above delivered, in which I think 
 no difficulty can occur. In the Gnomonical Projec- 
 tion, the plane projected on, is fuppofed to touch 
 the hemifphere to be projected, in its vertex-, and 
 the point of contact will be the center of projeclion. 
 ]Put if it be required to projeft upon any plane pa- 
 
 nlld
 
 40 GNOMONICAL PROJECTION 
 
 Fig. rallcl to this touching plane, the procefs will be no 
 
 42. way different, and is only taking a greater or lefTer 
 radius of projedion, according to the greater or 
 leffer diftance ; which is in effedt projefling a great- 
 er or leffer Iphere upon its touching plane. 
 
 When you have the fphsre to proje6t this way, 
 upon a given plane -, it will affift the imagination, 
 if you fuppofe yourfelf placed in the center of the 
 Iphere with your face towards the plane, whofe pofi- 
 tion is given ; and from thence projedling with 
 your eye, the circles of the fphere upon this plane. 
 
 PROP. IX. Prob. 
 
 43* 'To draw a great circle^ thro' a given -pointy and. 
 at a given difiancefrom the 'pole of proje5lion. 
 
 Rule. 
 Defcribe the circle ADB with the radius of pro- 
 jeftion, and thro' the given pomt P draw the right 
 line PCA, and CE perpendicular to it •, maice the 
 angle CAE = given diitance of the circle from C, 
 and thro' E defcribe the circle EFG, and thro' P 
 draw the line PK touching the circle in I, then is 
 PIK the circle required. 
 
 By the plain Scale. 
 With the tangent of the circle's diftance from 
 the pole of projection C, defcribe the circle EIF, 
 and draw PK to touch this circle; and PIK, is the 
 circle required. 
 
 PROP. X. Proh. 
 
 43. To draw a great circle perpendicular to a given 
 great circle^ which paffes thro' the pole of proje^ion \ 
 and at a given dijlance from that ^ole. 
 
 Rule. 
 Draw the primitive ADB. Let CI be the given 
 circle, draw CL perpendicular to CI, and make the 
 
 angle
 
 Sea. III. OF THE SPHERE. 41 
 
 angle CLI — the given dillance ; thro' 1 drawKPFis;. 
 parallel to CL for the circle required. 43. • 
 
 By the Scale. 
 ■ In the given circle CI, fet the tangent of the 
 given diftance, from C to I ; thro' I draw KP per- 
 pendicular to CI, then KP is the circle- required. 
 
 PROP. XI. Proh. 
 
 ^0 measure any part of a great circle ; or to fet any 44. 
 number of degrees thereon. 
 
 Rule, 
 
 Let EP be the great circle ; thro' C draw ID 
 perpendicular to EP, and C3 parallel to it. Let 
 EBD be a circle defcribed with the radius of pro- 
 jeftion CB, make I A zz IB; then A is the dividing - . 
 center of EP, confequently drawing AP, the •< 
 lAP IT meafure of the given arch IP. 
 . Or if the degrees be given, make, the <i lAP 
 zz thefe given degrees, which cuts off IP, the arch. , 
 correfpondent thereto. 
 
 By the Scale. 
 
 Draw ICD perpendicular to EP -, apply CI to 
 the tangents, and fet the femi-tangent of its com- 
 plement from C to A, gives the dividing center of 
 EP, &c. 
 
 PROP. XII. Proh. 
 
 To draw a great circle to make a given angle with ^ j , 
 a given gveat circle., at a given point ; cr to meafure 
 an angle made by two great circles. 
 
 Rule. 
 Let P be the given point, and PB the given great 
 circle. Draw thro' P, and C the center of projection, 
 the line PCG, to which from C draw CA perpen- 
 dicular,
 
 42 GNOMONICAL PROJECTION 
 
 Fig. dicular, and equal to tht radius of projeftion. Draw 
 51. PA and AG perpendicular to it, at G ere;5l BD per- 
 pendicular to GC, cutting PB in B -, draw AO bi- 
 fefting the angle CAP ; then at the point O, make 
 BOD — angle given, and from D draw the line 
 DP, then BPD is the angle required. 
 
 Or if the degrees in the angle BPD be required, 
 from the points B, D, drav; the lines BO, DO j 
 and the angle BOD is the meafure of BPD. 
 
 Cor. If an avgle he required to he made at the pole 
 or center of projctJiony equal to a given angle ; this is 
 no more than drawing two lines from the center mak- 
 ing the angle required. And if one great circle he to 
 he drawn -J- to another great circle^ it muji be drawn 
 thro' its pole. 
 
 PROP. XIII. Proh. 
 
 43' to project a leffer circle parallel to the primitive. 
 
 Rule. 
 
 With the radius of projeftion AC, and center 
 C, defcribe the primitive circle ADB, by Cor. Prop. 
 III. and draw ACB, and GCE perpendicular to it* 
 
 Set the circle's diftance from its pole from B to 
 H, and from H to D, and draw AFD. With 
 radius CE defcribe the circle EFG required. 
 
 By the Scale. 
 
 With the radius CE eciual to the tang-ent of the 
 circle's diftance from its pole, defcribe the circle 
 EFG, for the circle required. 
 
 PROP. XIV. ProK 
 
 48. 7'o drazv a lefjer circle perpendicular to the plane of 
 projection. 
 
 Rule. 
 Tliro' the center of projecaon C, draw its pa- 
 rallel great circle TI. At C make the angle ICN" 
 
 and
 
 Scd. III. OF THE SPHERE. 43 
 
 and TCO rz the given circle's diftance from its pa- Fig. 
 rallel great circle TI ; make CL equal radius of 48. 
 projeftion, and draw LM perpendicular to CL. 
 Set LM from C to V, or CM from C to F. Then 
 thro' the vertex V between the affymptotes CN, 
 CO defcribe the hyperbola WVK. Or to the fo- 
 cus F, and femi-tranfverfe CV^ defcribe the hyper- 
 bola i for the circle required. 
 
 Otherwife by Points, 
 
 Thro' the center of projeftion C draw the line of 
 meafures CF, and TCI perpendicular to it, draw 
 any number of right lines CV, DE, GH, IK &c. 
 and PQ, RS, TW, &c. perpendicular to TL And 
 by Prop. XI. make CV, DE, GH, &c. each e- 
 qual to the diftance of the given circle from its pa- 
 rallel great circle •, then all the points W, S, Q, V, 
 E, H, K, &c. joined by a regular curve will be 
 the reprefentation of the circle required. 
 
 Or thus. 
 
 Make the angle iak — diftance of the ^iven cir- 
 cle from its parallel great circle. Then thro' the 
 center of projection C, draw the great circle TCI 
 parallel to the circle given, upon v/hich ereft the 
 perpendicular CA r: radius of projeftion. Alfo 
 draw any number of right lines CV, DE, GH, IK, 
 &c. perpendicular to TI. Then take each of the 
 diftances from A to C, D, G, I, &c. and fct them 
 from a to <:, g, d, /, &c. and to ai draw the per- 
 pendiculars CV, de, gh, ik, &c. and make CV, 
 DE, GH, IK, &c. refpe(5fcively equal to cv, de, 
 gh^ ik. Sec. which gives the points V, E, H, K, 
 &c. after the fame manner on the other fide, find 
 the points Q, S, W, &c. then thro' all thefe points 
 W, S, Q, V, E, H, K, &c. draw a regular curve, 
 which will be an hyperbola reprefenting the cir- 
 
 cle given. 
 
 Bv
 
 44 GNOMONICAL PROJECTION 
 
 Fig- By the Scale, 
 
 ' Take the tangent of the circle's diftance from 
 its parallel great circle, and fct it from C (the cen- 
 ter of projedion) to V, and the fecant thereof from 
 C to F. Then with the femi-tranfverfe CV, and 
 focus F, defcribe the hyperbola WVHK. 
 
 P R O P. XV. Prob. 
 
 S'o proje£i any lejfer oiliq^iie. circle given. 
 
 Rule. 
 
 Draw the line of meafures dp, and at C the 
 ^^' center of projection draw CA -i- to dp and zz radius 
 of projection ; with the center A, defcribe the cir- 
 cle DCFG •, and draw RAE parallel to dp. Then 
 take the greateft and lead diftances of the circle 
 from the pole of projection and fet from C, to D 
 and F, for the circle DF -, and from A, the pro- 
 je6ting point, draw At/, and ADd, then df will 
 be the tranfverfe axis of the ellipfis. But if D 
 fall beyond the line RE, as at G, then draw a line 
 from G backward thro' A to D, and then df is 
 the tranfverfe of an hyperbola. But if the point 
 D fall in the line RE as at E, then the line AE no 
 where meets the line of meafures, and the projec- 
 tion of E is at an infinite diftance, and then the cir- 
 cle will be projedted into a parabola whofe vertex 
 is/. Laftly, bifecSt ^in H the center, and for the 
 ellipfis take half the difference of the lines Ady 
 A/, and fet from H to K for the focus. But for 
 the hyperbola take half the fum of Ad, A/, and fet 
 from H to the focus k of the hyperbola Then with 
 the tranfverfe df and focus K or ^ defcribe the el- 
 lipfis dMf or the hyperbola/;;;. For the projec- 
 ;i9n of the circle given. 
 
 But for the parabola make EQ^zi: F/, and draw 
 fn -»- AQjand fet -^;/Q^from/toK the focus. Then 
 
 with
 
 Sea. III. OFTHESPKERE. 45 
 
 with the vertex /and focus k defcribe the parabola Fi^. 
 /w, for the projedlion of the given circle FE. 
 
 Otherwife hy Points. 
 
 Thro' the center of projedion C, draw the line .g, 
 of meafures CF, pafllng thro' the pole P (if P is 
 given ; but if not, find it, by fetting oft CP zz the 
 diftance of that pole, from the center of projedlion, 
 by Prop. XI.) then fet off PD, PF equal to the 
 given diftance from its pole, by Prop. XI. Thro' 
 P draw a fufficient number of right lines, L>, M^., 
 N», Oo, Rr, Sj, &c. which will all reprefent great 
 circles. Find the dividing centers of each of thefe 
 lines ; and by Prop. Xf. fet off upon each of them 
 from P, the given diftance of the circle from its 
 pole, as PL, Pa, PiM, P;^., &c. and thro' ail the 
 points L, M, D, O, R, &c. draw a curve line, for 
 the circle required. 
 
 Or thus. ' 
 
 Draw the line of meafures PCG, and by Prop. 49< 
 XI. make CG — the diftance of the parallel great 
 circle from the pole of projeftion, and draw AGK 
 perpendicular to it, which will reprefent a great 
 ciicle v/hofe pole is P. Draw any number of right 
 lines thro' P to AK, as AP, BP, HP, &c. and by 
 Prop. XI. fet off from AK the parts AL, BM, HG, 
 &c. each equal to the circle's diftance from its pa- 
 rallel great circle. Then all the points L,, M, D, 
 O, &c. being joined by a regular curve, will repre- 
 fent the parallel circle required. 
 
 Or thus. 
 
 -Thro' the center of projeftion C draw the line of 40. 
 meafures DGF, and the radius of projedtton CW 
 perpendicular to it, and AGK -^ GC, for a great 
 circle whole pole is P. Draw wp zz WP, and wa 
 ■^ to it, draw any number of right lines, AP, BP, 
 GP, he and make /)?, pb, pc, &c. Z2 FG, PB, PA, 
 
 D 2cc.
 
 46 GNDMONICAL PROJECTION 
 
 fig- &c. alio make the <C pwl and pwx — the circle's 
 
 49. diftance from its pole P (or awl — the diftance from 
 
 its parallel great circle) •, and upon PG, PB, PA, 
 
 &c. make PD, PM, PL, &c. —pd, pm, pl^ &c. re- 
 
 Ipeftively. 
 
 Or make GD, BM, AL, &c. —gd, hm, al, he. 
 After the fame manner, find the points O, R, &c. 
 and thro' all the points R, O, D, M, L, &c. draw 
 a regular curve, making no angles, which will re- 
 prefent the parallel required. Likewife where any 
 line ap cuts wx^ that diftance from p will give the 
 point A, or is — Pa-, and lb of any other of th© 
 lines />/), gp^ Sec. 
 
 Tbe reafon of this procefs 'will be plain., if you fup- 
 pofe the points p, "W applied to P, W j and g., b., ^, ^c. 
 fucceffively to G, B, A, 6cc. for then d, m, /, will fall 
 upon D, M, L, &c. 
 
 By the Scale. 
 ^-. Take the tangents of the circle's neareft and fur- 
 theft diftance from the pole of projeftion, and ^ti 
 from C to / and d, gives the vertices, and bifed df 
 in H •, then take half the difference, or half the fum, 
 of the fecants of the greateft and leaft diftances from 
 the pole of projedion, and fet from H, to K or 
 k for the focus of the ellipfis or hyperbola, which 
 may then be defcribed. 
 
 49. Cor. If the curve be required to pnfs thro'' agi'ven 
 point S •, 7nenfure PS by Prop. XI, and then the curvf 
 may be drawn by this Problem. 
 
 PROP. XVI. Prob. 
 
 47. To find the pole of any circle in the proJe5lion^ DMF. 
 
 Rule. 
 From tlie center of projedion C, draw the radius 
 of projeclion CA perpendicular to the line of mea- 
 sures t
 
 Sea. lU. OF THE SPHERE. 47 
 
 fiires DF. And to A the projedling point, draw Fig. 
 DA, FA, and bifeft the angle DAF by the line AP, 47. 
 then P is the poie. But if the curve be an hyper- 
 bola, as />«, fig. 45, you muft produce dA, and bi- 
 fedt the angle /AG. And in a parabola, where the 
 point <^ is at an infinite diilance, bifeft the angle/AE. 
 Or thus ; Drawing CA perpendicular to DC, draw 
 DA, and make the angle DAP :=: the circle's dii- 
 tance from its pole, gives the pole P. 
 
 By the Scale. 
 
 Draw the radius of projedlion CA -^ to the 
 line of meafures DF. -'^'Pply CD CF to the tan- 
 gents, vnd fct the tangent of half the difference of 
 their degrees from C to P, if D, F lye on contrary 
 fides of C ; but half the fum if on the fame fide, 
 gives P the pole. 
 
 Or thus i By Prop. XI. fct off from D to P, the 
 circle's diilance from its pole, gives the pole P. 
 
 Cor. If it be a great circle ^j- EG ; draw the live 4.6. 
 of meafures GC, and CX -^ to it, and equal to the 
 radius of -proje^ion^ make GAP a right angle .^ and 
 P is the -pole. 
 
 PROP. XVII. Prch. 
 
 "To meafure any arch of a Icffer circle \ or to fet ary 
 number of degrees thereon. 
 
 Rule. 
 
 Let Fw be the given circle. From the center of .5^ 
 projeftion C, draw CA perpendicular to the line 
 of meafures GH. To P the pole of the given cir- 
 cle draw AP, and AG bifefting the angle CAP. 
 And draw AD perpendicular to AG. Defcribe 
 the circle GIH [by Prop. Xlll.) as far from the pole 
 itif proie<'l:ion C, as the given circle is from its pole 
 i*.- And thro' any given point ?i in the circle Yn., . 
 
 D 2 draw
 
 4^ GNOMONICAL PROJECTION 
 
 Figv draw Dfil, gives H/ the number of degrees zz F«. 
 4.^. Or the degrees being given and let from H to /, the 
 line D/ cuts off Fn equal thereto. 
 
 Or thus \ AG being drawn as before, erefl OS 
 perpendicular to CO j thro' the given point n draw 
 P;/ cutting OS in Q, then thro' Q draw C/, and the 
 angle QCP is — F;;. Or making QCP = the de- 
 grees given, draw PQ^;, and arch Fn — thefe de- 
 grees. 
 
 Or thus; AO, AP, being drawn as before, draw 
 AG perpendicular to AP, and GB perpendicular 
 •to GC. Thro' the given point n draw PB cutting 
 GB in B, and draw OB, then the <: GOB — arch 
 F?i. Or making <i GOB — the given degrees ; draw 
 PB, and it cuts off Fn rz the degrees given. 
 
 By the Scale, 
 
 Let C be the center of projeflion, P the pole of 
 the given circle. Apply CP to the tangents, and 
 fet the tangent of its half from C to O, and the co- 
 tangent of its half from C to D ; with radius CG iz 
 tangent of the degrees in FP the given circle's dif- 
 tance from its pole, defcribe the circle GSH. Then 
 D/ drawn thro' « or /, cuts off H/ — Fn. 
 
 Or thus i O being found as before, ere6l OS per- 
 pendicular to CO •, thro' the given point n draw 
 PQ/;, and <lQCH^Fn. 
 
 Or thus J Apply CP to the tangents, and fet the 
 co-tangcnt tiicreof from C to G. Erecl GB per- 
 pendicular to GC. Thro' n draw P»B, and draw 
 i^.O i then < GOB - Fn. 
 
 4S. Cor. If the lejfer circle he perpendicular to the plain 
 of p-GJc5lion OS VHK. T'oh have no more to do but 
 to dra'-jj the perpendiculars VC, FIG, to its par-alkl 
 great circle CL Then CG {meafured by Prop. XL) 
 iv'/// be equ^il to VH ; cr the degrees fet from C to G, 
 cuts off Vll equal thereto. 
 
 S C H O-r
 
 Sea. III. OF THE SPHERE. 49 
 
 Fig. 
 Scholium. 
 
 This fort of proje6lion is little ufed, by reafonof 48. 
 fevcral of the circles of the fphere fall in ellipfes 
 and hyperbolas, which are very difficult to defcribe. 
 Notwithftanding it is very convenient for folving 
 fome Problems of the fphere, becaufe all the great 
 circles are proiefted into right lines. And this forr, 
 or the Gnomonic Projection is the very foundation 
 of all dialling. For if the fphere be projeded on 
 any plane, and upon that fide of it on which the 
 fun is to fhine j and the projefted pole be made 
 the center of the dial, and the axis of tlitr globe 
 the Stile or Gnomon, and the radius of projection its 
 height ; you will have a dial drawn vv'ith all its fur- 
 niture. Upon this account it deferves to be more 
 taken notice of, than at prefent it is. I have in the 
 foregoing proportions given, I think, all the fun- 
 xdamental principles of this kind of projection, ha- 
 ving met with little or nothing done upon this fub- 
 jed; before. 
 
 GENERAL PROBLEM. 
 
 5"^ projeci the fphere upon any given plane. 
 
 Before you can projeCt the fphere upon any plane, 
 you m,uft have a perfeCt knowledge of all its cir- 
 cles, and their pofitions in refpect of one another •, 
 the diitances of the lefier circles from their poles, 
 and from their parallel great circles ; the angles 
 made by great circles, or their Inclinations, to one 
 another, particularly to the primitive circle, on 
 whofe plane 'or a parallel thereto) you are about ta 
 project the fphere. Then after the primitive cir- 
 cle is defcribed ; you mud defcribe all other circles, 
 concerned in the Problem, according to tiie rules 
 of that fort of Projection, you are going to ufe -, 
 
 D 3 and
 
 50 GNOMONICAL PROJECTION 
 
 VW. and the interfcflion of thefe circles will determine 
 the Problem. 
 
 And note, that the Projedion of the concave fide 
 of the fphere is more fit for aftronomical purpofes ; 
 for in looking at the heavens, we view the conca- 
 vity. Bur it is better to projeft the convex hemif- 
 phere in geography, becaufe we fee the convex fide 
 only. 
 
 The principal Points^ Angles and Circles of the Sphere 
 are as follows. 
 
 I. Points. 
 i. Zenith is the point over our heads, Z. 
 ' 2. Nadir is the point under our feet, N. 
 . 3. P(9/f^ of the world are 2 points, round which 
 the diurnal revolution is performed, P the north 
 pole, p the fouth pole. A line drawn through the 
 poles, is called the Jxis of the world, as Fp. 
 
 4. The Center of the earth or of tlie heavens, C. 
 
 5. Equinotlial Points^ are the points of interfec- 
 tion of the Equator and Ecliptic, t , —• 
 
 6. Solftitial Points., are the beginning of Cancer 
 and Capricorn, 5c, \y. 
 
 II. Great Circles. 
 
 1. Equino5fialj is a circle 90 degrees diftant from 
 the poles of the v/orld, as £Q. 
 
 2. Meridians., or hour Circles ; are circles palTing 
 thro' the poles of the world, as Pop, PE/*, <5cc. 
 
 ^. Sclfiitial Colure, is a meridian paffing thro' the 
 folftitial points, as Ps/>. 
 
 4. Eqtrinc^ial Colure, is a meridian pafling thro* 
 the equinoftial points, P Cp. 
 
 5. Ecliptic is the circle thro' which the fun feems 
 to move in a year, s 'jy •, it cuts the equino6lial at 
 an aiigle of 23° 30', in paffing thro' the equinodial 
 points. In this are reckoned the 12 Sines, Tj t:» 
 T.y S, a> '^X, :^, ni, /^, v/, c:r, X. 
 
 6. Ho~
 
 
 n/^ ' 
 
 
 \, 
 
 / 
 
 /\x \,^^^ 
 
 \/ \ 
 
 / 
 
 4 \v 
 
 /V 
 
 1 
 
 
 E T^ 
 
 / Xi. 
 
 
 A 
 
 / 
 
 
 
 \ 
 
 \X^ 
 
 ^d 
 
 c
 
 X.^.jo.
 
 r
 
 Sea. III. OF THE SPHERE. 5^, 
 
 6. Horizon^ is. a circle dividing the upper from Fig. 
 the lower hemifphere, as HO, being 90^ diitant 52. 
 from the Zenith and Nadir. 53. 
 
 7. Vertical Circles^ are circles pafling thro' the ^j. 
 Zenith and Nadir, Z N. 
 
 8. Circles of Longitude in the heavens, pafs thro* 
 the poles of the ecliptic and cut- it at right angles. 
 
 9. Meridian of a Place, is that Meridian which. 
 pafTes thro' the Zenith, as PZH. 
 
 10. Prime Vertical, is that which pafTes thro' the 
 «afl: and weft points of the horizon. 
 
 III. Leffer circles. 
 
 1. Parallels of Latitude are parallel to the equi- 
 nodlial on the earth, parallels of altitude are parallel 
 to the horizon, parallels of declination are parallel 
 to the equinodial in the heavens. 
 
 2. Tropics, are 2 circles diftant 23= 3.0' from the^. 
 equinodtial, the tropic of Cancer towards the northj. 
 the tropic of Capricorn towards the fouth. 
 
 3. Polar Circles, are diftant 23° 30' from the 
 poles of the w^orld, the Arctic circle towards the 
 north, the Antarctic towards the fouth. 
 
 IV. Angles and Arches of Circles, 
 
 1 . Sun^s (or Starts) Altitude, is an arch of the A- 
 zimuth between the iiin and horizon, as o B. 
 
 2. Amplitude is an arch of the horizon, between 
 fun-rifing and the eaft, or llm-letting and the weft. 
 
 3. Aximuih, is an arch of the horizon between 
 the fun's Azimuth circle, and the north or fouth,. 
 as HB, or OB -, or it is tlie angle at the zenith, 
 HZBor OZB. 
 
 4. Rigpjt Afcenfion is aa arch of the eq\iator be- 
 tween the fun's meridian, and the firft point of 
 Aries, as T K. 
 
 5. Afccnfional Difference is an arch of the equi- 
 noctial, between the fun's meridian, and that point 
 
 D 4 oT
 
 5* GNOMONICAL PROJECTION 
 
 Mg. of the equiriodlial that rifes with him, or it is the 
 52. angle at the pole between the fun's and the fix o'clock 
 5^. meridian. 
 
 §^. 6. Oblique Afcenfion or Defcenfion^ is the fum or 
 difference ot the rigrht afcenfion and the afcenfional 
 
 o 
 
 difference. 
 
 7. Stiffs Longitude, is an arch of the ecliptic, be- 
 tween the fun and firft part of Aries, as x O- 
 
 8. Declination is an arch of the meridian, between 
 the cquinodlial and the fun, as OK. 
 
 9. Latitude of a Star, is an arch of a circle of 
 longitude between the ftar and ecliptic. 
 
 10. Latitude cf a Plane, in an arch of the meri- 
 dian between the cquinodlial and the place. 
 
 1 1. Longitude of a place on the earth is an arch, 
 of the equino6lial, between the firft meridian (Ifle 
 of Ferro), and the meridian of the place. And 
 diff. longitude, is an arch of the equator, between the 
 rueridians of the two places, or the angle at the pole. 
 
 12. Hour of the Day, is an arch of the equinoc- 
 \ tial, between tiie meridian of the place and the fun's 
 
 meridian, as EK ; or it is the angle they make at 
 the pole, as EPO. 
 
 Example I. 
 'To proje^l the fphere upon the plane of the meridian^ 
 for May 12, 1767. Latitude 54'' -^ north, at a quar- 
 ter pajl ^ o'clock before noon. 
 
 I. By the Orthographic Proje^ion. 
 1 2. Here we will projecSl the convex fide of the eaft- 
 crn hemifphere. With the chord of 60' degrees 
 defcribe the primitive circle or meridiiii of the place 
 HZON. Thro' the center C draw the horizon HO ; 
 let the latitude 544 from O to P and from H top, 
 and draw Vp the 6 o'clock meridian. Thro' C draw 
 I'.Q perpendicular to Vp for the equinoctial. 
 Make ED, Qd 18= 5' the declination May 12, 
 .ijid draw Y)d the fun's parallel for that day. By- 
 Prop.
 
 ^■F-^^
 
 c
 
 Sea. III. OF THE SPHERE. 53 
 
 Prop. XI. make O G (3 4- hours or) 48° 45' the Fig. 
 fun's diftance from the hour of 6, then O is the 52. 
 fun's place. Thro' O by Prop. V. draw AL pa- 
 rallel to H for the fun's parallel of altitude. By 
 Prop. VII. draw the meridian P ©/> and the azimuth 
 Z O N. Alfo the ecliptic will be an ellipfis pafling 
 thro' 0, which cannot conveniently be drawn in 
 this projeftion. Alfo draw the parallel Sj i 8^ be- 
 low the horizon, and where it interfedts Dd is the 
 point of day break, if there is any. Now the fun 
 is at <^ at 1 2 o'clock at night, and rifes at R, at 6 
 o'clock is at G, due eaft at F, at a quarter paft 
 5, and is at D in the meridian at 1 2 o'clock. 
 
 Draw GI parallel to HO. Then GR meafured 
 by Prop. X. is 27° 14', and turned into time fallow- 
 ing 15 degrees for an hour) fliows how long the fun 
 rifes before 6, to be i** 49'" ; GI meafured by Prop. 
 X. gives the azimuth at 6, 79° 1 6'. CR by Cor^ 
 Prop. X. gives the amplitude 32° 19', and CF gives 
 his altitude when eaft 22° 25'. FG 13° 28' (turned 
 into time) is 54"", and Ihews how long after 6 he 
 is due eaft. lO is his altitude at 6, 14° 38'. AH 
 41° ^^' is his altitude at ©, or a quarter paft 9; 
 and ©L meafured by Prop. X. is his azimuth 
 from the north at the fame time, 122° 40'. And 
 thus the place of the moon or a ftar being given, it 
 may be put into the projeAion, as at sjc. And i» 
 altitude, azimuth, amplitude, time of rifmg, &c. 
 may all be found, as before for the fun. 
 
 II. Stereographically. ^ 
 
 To projeft the fphere on the plane of the meri- ^3. 
 dian,the projeding point in the weftern point of the 
 horizon •, with cord of 60, draw the primitive cir- 
 cle HZON, and tliro' C draw HO for the hori- 
 zon, and ZN perpendicular thereto for the prime 
 vertical. Set the latitude from O to P, and from 
 H to /», and draw Pp the 6 o'clock meridian, and 
 
 EQ
 
 54 GNOMONICAL PROJECTION 
 
 Ficr. EQ^ perpendicular thereto for the equino£i:iaI. 
 
 5^. Make ED, Qi the declination, and by Prop. XII. 
 draw DGdy the fun's parallel for the day. Draw 
 the meridian Fop by Prop. XVII. making an an- 
 gle of 41° 15' with the primitive, to interied: the 
 fun's parallel in G , the fun's place at g^ ■^. Thro* 
 G , by Prop. XII. draw the parallel of altitude 
 A G L ; thro' G draw, by Prop. XVII. the azi- 
 muth ZgN. And by Prop. XII. draw the paral- 
 lel Ssd 18° below the horizon, if it cut R^, gives 
 the point of day break. And thro' G draw the 
 parallel of altitude GI. Laftly, by Prop. XX. 
 thro' O draw the great circle v"0~ cutting the 
 equinoctial EQ at an angle of 23° : 30', and this 
 is the ecliptic, t the firll point of Aries, and ;2: 
 that of Libra. 
 
 This done, ^R meafured by Prop. XXIII. is 62* 
 46', fhows the time of fun rifing ; CR by Prop. 
 XXII. is the amplitude 32° 19'. GI 79° 16' by 
 Prop. XXIII. the fun's azimuth at 6. lO )4'> 38' 
 his altitude at 6. CF 22° 25' by Prop. XXII. his 
 altitude when eaft. GF 13" 28' the time when he 
 is due eaft. oB 41° 53' by Prop. XXII. his alti- 
 tude at a quarter paft 9 ; the <:oZP 122° 40' by 
 Prop. XXIV. his azimuth at that time. Alfo TO, 
 by Ir'rop.XXIl. is his longitude 51^7'. 'V'K his right 
 afcenfion, 48^ 40'. 
 
 And the place of the moon or a ftar being given, 
 it may be put into the fcheme as at * ; and its 
 time of rifing, amplitude, azimuth, &c. found as 
 before. 
 
 III. Gnomonically. 
 
 54. To projed the eaftern hemifphere upon a plane 
 parallel to the meridian. About the center of pro- 
 jcdion C defcribe the circle HON with the tangent 
 o\ 4.5 the radius of proiedion, for the primitive, 
 'lliro' C draw the horizon MO, and the prime ver- 
 tical
 
 Sea. 111. OF THE SPHERE. 55 
 
 tical ZN perpendicular thereto. Set the latitude Fig. 
 54 4- from H to «, and draw the 6 o'clock meri- 54. 
 dian P/>, and the equino6tial EQ^perpendicular to 
 it. Set the tangent of 48° 45' (equal to ^~ hours) 
 from C to E, and by Prop. X. draw the meridian 
 EL parallel to P/>. Make Ee zr E^, and <Z Ee Q 
 = 18° 5' the fun's declination, then by Prop. 
 XI. O is the fun's place. Thro' O draw the hy- 
 perbola D0^ (by Prop. XIV.) for the fun's paral- 
 lel of declination -, and draw OB perpendicular to 
 HO, for his azimuth circle. And draw GI per- 
 pendicular to HO, and RM, FT, || Pp. Alio the 
 ecliptic is a right line pafllng thro' 0, and cutting 
 EQ^at an angle of 23° 30', which is difficult to 
 draw in this proje6lion. 
 
 Alfo by Prop. XIV. Draw the parallel Sj 18" 
 below the horizon, and if it interfeds Dd^ it gives 
 the point of fun rife. 
 
 Then if by Prop. XVII. or XI. you meafure GR 
 or rather CM, 27° 14', you have the time of fun 
 rifing i GF or CT 1 3=^ 28 , the time when he is due 
 call. Alfo by Prop. XI. if you meafure CR you 
 have the am.plitude 32° 19'. CI the comp. of his 
 azimuth at fix, 10° 44'. IG by Prop. XII. his al- 
 titude at 6, 14° 38'. CF his altitude when eaft, 
 22° 25'. And by Prop. XL ©B n: 41° ^^\ his 
 altitude a quarter paft 9. CB the complement of 
 his azimuth at that time, 32° 40'. 
 
 And the place of the moon or a ftar being given, 
 its place in thv projedion may be determined as be- 
 fore, and all the requifites found. 
 
 Ex. 2 . 
 
 To proje^ the fphere upon the plane of the foljii- 
 tlal colure for latitude c^\~ IL May 23, i-"]^^-, ^t 
 10 o^clcck in the mcrnin^. 
 
 Stere-jgra-
 
 ^6 GNOMONICAL PROJECTION 
 
 55' 
 
 Stereogrophically. 
 The proje6lion of the weftern hemifpherc, thcr 
 firft point of Libra, the projefling point. Defcribc 
 the folftitial colure PEpQ,- and the equinodtial co- 
 lure \?p perpendicular to it; and thro' C draw the 
 equinoftial EQ^perpendicular to Vp. Set 23° 30 
 from E to s, and from Q^to Vf, and draw the e- 
 cliptic s V?. Set the fun's longitude 61° 42' from 
 C to G , and thro* draw POKp for the 10 o'clock 
 meridian. Make KA (two hours or) 30°, and draw 
 PA/> for the meridian of the place. Set the lati- 
 tude of the place c^\\ from A to Z, and Z is the 
 zenith. About the pole Z defcribe the great circle 
 BHS for the horizon of the place. Thro' Z and 
 G draw an azimuth circle ZOB. 
 
 Then you have 0K the fun's declination 20° 33'. 
 CK his right afcenfion 59^ 35'. ©B his altitude 
 at 10 o'clock 49^ 10' J the < AZo or PZ© his azi- 
 muth at 10 zr HB, 45° 44'. H the fouth point' 
 of the horizon. I the point of the ecliptic that is 
 in the meridian. T the point of the ecliptic that 
 is fettuig in the horizon. 
 
 Example. 3. 
 
 ^0 projeH the fphere on the plane of the horizofty 
 Lai. 35I, N. July 31, 1767, ai 10 o'clock. 
 
 Gnomonically. 
 
 ■ 5_ To projccl the upper hemifphere on a plane pa- 
 rallel to the horizon. With the radius of projec- 
 tion and center C, defcribe the primitive circle 
 ADB. Thro' C draw the meridian PE, and AS 
 perpendicular to it for the prime vertical. Set 
 off CP 35; the latitude and P is the N. Pole, and 
 perpendicular to CP draw Vp the 6 o'clock meri- 
 dian. Set the complement of the latitude from C 
 ^ti> E •, and draw EQ perpendicular to CE for the 
 
 equinoc-
 
 Ict^t
 
 r
 
 Sea. III. OF THE SPHRRE. 57 
 
 equino6lial. Make EB 30° (or 2 hours) and draw Fig. 
 the 10 o'clock meridian PB. Set the fun's dechna- ^t>. 
 tion 18° 27' from B to 0. And O is the place of 
 the fun at 10 o'clock. Thro' O draw the azimuth 
 circle CQ^i likewife thro' 0, a parallel to the equi- 
 nodtial EQ^may eafily be defcribed by Prop. XV. 
 for the fun's parallel that day. 
 
 Then C meafured by Prop. XL is 31° 30' the 
 complement of the altitude. And the angle ECO 
 meafured by Cor. Prop. XII. is his azimuth, ds". 10. 
 
 Scholium. 
 After this manner may any Problems of the 
 Sphere be folved by any of thefe Projed:ions, or 
 upon any planes, but upon fome more com mod i- 
 oufly than upon others. And if in a fpherical tri- 
 angle any fides or angles be required, they may be 
 projected according to what is given therein, accord- 
 ing to any of thefe kinds of projedtion before de- 
 livered ; and it will be moil eafily done, when you. 
 chufe fuch a plane to projeft on, that fome given 
 fide may be in the primitive, or a given angle at 
 the center ; and then you need draw no more lines 
 or circles than what are immediately concerned 
 in that Problem. But always chufe fuch a plane 
 to projcsfl on, where the lines and circles are moll: 
 eafily drawn, and fo that none of them run out of 
 the fcheme. 
 
 F I N I S..
 
 ERRATA. 
 
 pag 
 
 line 
 
 5 
 8 
 
 1 
 l8 
 
 9 
 
 3b 
 
 »3 
 
 17 
 
 2b 
 
 
 21 
 
 15 
 
 i6 
 
 i8 
 
 20 
 21 
 
 25 
 29 
 
 30 
 
 33 
 36 
 
 43 
 
 44 
 
 5 
 
 4 
 
 7 
 
 »5 
 
 9b 
 
 6b 
 
 2 
 
 nb 
 
 «7 
 6b 
 2b 
 2b 
 
 2 
 20 
 
 b /ignifies reckon from the bottom. 
 
 read 
 fig. 2. 
 
 %-5- 
 
 off the fines, 
 
 fig. 12. 
 
 the 2 laft lines (hould be indented and reman. 
 C/A -}- CA/' ; 
 
 the 3d line fhould be indented, and the 3 follow- 
 ing lines roman. 
 ECI =: CAD 
 if, /, q, be 
 projeftion C, 
 OG + OD . 
 
 points A, B, G, 
 interfedion /, 
 gCt required, 
 pole p, draw 
 of this circle 
 TV, and 
 at s cut 
 to oS ; 
 CL — radius 
 A to d'.
 
 THE 
 
 LAWS 
 
 O F 
 
 Centripetal and Centrifugal Force* 
 
 SHEWING, 
 
 The Motion of Bodies in Circular Orbits, and 
 in the Conic Sedions, and other Curves. 
 
 And explaining the perturbating Force of a third 
 Body. With many other Things of like Nature. 
 
 Being a Work preparatory to Astronomy, and 
 the very Bafis thereof. And abfolutely neceflary 
 to be known by all fuch as defire to be Profi- 
 cients in that Science. 
 
 Solis uti varios curfus^ lunaque tneatus 
 Nofcere poffemuSf qua vis, k^ caufa cieret 
 
 LucRET. Lib. V.
 
 [ > ] 
 
 THE 
 
 PREFACE. 
 
 TN the following Treat'ife^ I have explained and detnonjlra^ 
 * ted the Laws of Centripetel Forces ; a doSfrine upon 
 which all AJlronomy is grounded ; and without the knowledge 
 of which, no rational account can be given of the motions 
 of any of the celefiial bodies, as the Comets, the Planets, and' 
 their Satellites. From the/e laws are derived the caufes of 
 the various feeming irregularities obferved in their motions ; 
 fuch as their accelerations and retardations, their approach- 
 ing to, and receding from the center of force ; irregularity, 
 enly in appearance ; but in reality, thefe motions are truly 
 regular and conformable to the ejlablijhed laws of Nature. 
 From this foundation we trace the way or path of all the 
 planets, and difcover the origin and fpring of all the celef- 
 tial motions, and clearly underjiand and account for all the 
 phanomena thence arifing. 
 
 In the fir/i feSlion, you have the Centripetal Forces of bo- 
 dies revolving in circles ; their velocities, periodic tifnes, 
 and dijlances cojnpared together ; their relations and propor- 
 tions to each other ; and that when they either revolve about 
 the fame center, or about different ones. The different mo- 
 tions caufed by different forces, or by different central at- 
 tracting bodies, are here Jhewn. We have given likewife 
 the periodic time of a fmple pendulu?n revolving with a coni- 
 cal motion ; and alfo the center of Turhination, and the pe- 
 riodic time of a compound pendulum, or a fyftem of bodies, 
 revolving with a conical motion \ as properly belonging ta 
 the doSirine of Centripetal Forces. 
 
 In the fecond feSlion we have Jhewn the motion of bodies 
 in the ElUpJis, Hyperbola, and Parabola ; and in other 
 Curves. The proportion of the Centripetal Forces, and ve- 
 locities in different parts of the fame Curve. The law of 
 Centripetal Force to defcribe a given Curve, and the vclo- 
 A 2 city.-
 
 11 
 
 The PREFACE* 
 
 city in any point of it ; and more particularly with refpeSl 
 to that law of Centripetal Force that is reciprocally as the 
 fquare of the dijiance ; zuhich is the grand law of Nature in 
 regard to the action of bodies upon one another at a dif- 
 tancc ; and according to this law^ is Jhewn the motion of bo- 
 dies round one another^ and round their common center of 
 gravity^ and the orbits they will defcribe. 
 
 In the third f cation we hai'c given the dijlurbing or per- 
 turbating force of a third bo dy^ a5iing upon two others that 
 revolve round one another. From thefe principles are de^ 
 duced the errors caufed in the motion of a Satellite^ moving 
 round its primary planet. Towards the end, are feveral 
 propoftions, by means whereof, the motion of the Nodes, and 
 variation of inclination of a Satellited orbit, and fuch like 
 thijigs may be computed. As thefe things are all laid down 
 for the fake of undcrjlanding our own Syjlem, I have infert- 
 cd fomc fc%u things, by way of illujlration of the rules, in 
 regard to the Moon and Jupiter. But as to the Moon, there 
 are fonie things fo very intricate, and require fuch long and 
 tedious calculations, as would require a volume of the?nfelves ; 
 fo that the fmall room I am confined to cannot admit of them ; 
 and few xuould trouble themfelves to read them, if they were 
 there. This laji fefiion concludes with a few things of am- 
 ther kind, but depending on the principles of Centripetal 
 Fcrces. 
 
 Several of thefe things about Centripetal Forces are calcu- 
 lated by the method of Fluxions ; and cannot eafily be done 
 any other way ; and mojt of them taken from 7ny book of 
 Fluxions. And feveral other things relating to Centripetal 
 Forces, you tuili alfo find in that book ; being forry to trou- 
 ble the reader too much with repeating what I have written 
 and pnblifijcd elfeivhere. 
 
 W. Emerfon.
 
 [ I ] 
 
 »i—ii—w BWiimm p Miw«»iiiiiii I iiitiBiri iwimii ■ « 
 
 THE 
 
 LAWS 
 
 O F 
 
 Centripetal and Certrifugal Force. 
 
 DEFINITIONS. 
 
 D E F. I. 
 
 ^H E center of attraBion^ is the point tov/ards 
 which any body is attra6ted or impelled, 
 
 D E F. II. 
 
 Centripetal force, is that force by which a body Is 
 impelled to a certain point, as a center. Here all 
 the particles of the body are equally adted on by 
 the force. 
 
 D E F.. III. 
 Centrifugal force, is the refiftance a moving body 
 makes to prevent its being turned out of its dired: 
 courfe. This is oppofite and equal to the centri- 
 petal force ; for adlion and re-aftion are equal and 
 contrary. 
 
 D E F. IV. 
 Angular velocity, is the quantity of the angle a 
 body defcribes in a given tim_e, about a certain 
 point, as a center. Apparent velocity is the fame 
 thing. 
 
 D E F. V. 
 Periodical time, is the time of revolution of a bo- 
 dv round a center. 
 
 B SECT,
 
 [ 2 ] 
 
 SECT. I. 
 
 *fhe ^notion of bodies in Circular Orbits. 
 
 PROP. I. 
 
 Fig» ^he centripetal forces^ wherehy equal bodies at equal 
 dijlances from the centers of force, are draivn to- 
 wards thefe cotters ; are as the quantities of matter 
 in the central bodies. 
 
 For fince all attraftion is made towards bodies, 
 every part of the attratling body muft contribute 
 its fhare in that efFeft. Therefore a body twice as 
 great will attraft the fame body twice as much ; 
 and one thrice as great, thrice as much, and fo on. 
 T4ierefore the attraction of the central body j that 
 is, the centripetal force, is as the quantity of mat- 
 ter in the attracting or central body. 
 
 Cor. I. Any body whether great or little^ placed 
 at the fame difiance, is attracted thro' equal f^ aces in 
 the fame time, by the central body. 
 
 For tho' a body twice or thrice as great as ano- 
 ther, is drawn with twice or thrice the force ; yet 
 it will acquire no greater velocity, nor pafs thro' a 
 greater fpace. For (Prop. V. Cor. 2. Mechan.) 
 the velocity generated in a given time, is as the 
 force dire<5tly, and quantity of matter reciprocally; 
 and the force, which is the weight of the body, 
 being as the quantity of matter ; therefore the ve- 
 locity generated is as the quantity of matter di- 
 reftly, and quantity of matter reciprocally, and 
 therefore is a given quantity. 
 
 Cor.
 
 Seel. I. CENTRIPETAL FORCES. 5 
 
 Cor. 2. therefore the centripetal forccy or force Fig. 
 towards the center^ is not to he meafured by the quan- 
 tity of the falling body ; but by the fpace it falls thro* 
 in a given time. And therefore it is fometimes called 
 an accelerative force. 
 
 PROP. II. 
 
 If a body revolves in a circle^ and is retained in it, 1 1 
 by a centripetal force.^ tending to the center of it ; -put 
 R zz radius of the circle or orbit defcrihed, AC, 
 
 F rz abfolute force^ at the dijiance R. 
 
 s z= the fpace ^ a falling body could defcend thro\ by 
 the force at A, and 
 
 t 1Z. time of the defcent. * 
 
 TT 1= 3-1416. 
 'Then its periodic time, or the time of one revolution 
 2R 
 
 njoill be TTt v/ 
 
 ^ s 
 
 And the velocity, or fpace it defcribes in the time /, 
 isjill be ^yiKs. 
 
 For let AB be a tangent to the circle at A -, take 
 AF an infinitely fmall arch, and draw FB perp. to 
 AB, and FD perp. to the radius AC. Let the 
 body defcend thro' the infinitely fmall hight AD or 
 BF, by the centripetal force in the time i. Now 
 that the body may be kept in the circular orbit 
 AFE, it ought to defcribe the arch AF in the 
 fame time i. The circumference of the circle AE 
 
 is 27rR, and the arch AF ir v/^R X AD. 
 
 By the laws of falling bodies ^/s : t : : v^AD ; 
 
 AD 
 
 / \/ ~ time of moving thro' AD or AF. And 
 
 by uniform motion, as AF, to the time of its de- 
 
 fcription : : circumference AFEA, to the time of 
 
 B 2 one
 
 4 CENTRIPETAL FORCES. 
 
 Fig. . ^ AD 
 
 J. one revolution ; that is, v/^R x AD : i */ — : : 
 
 _ . J. . 2/7rR 2R 
 
 27rR : periodic time i= —rrr ~ tt / ./ 
 
 ^ y/2KS ^ s 
 
 AD 
 Alfo by the laws of uniform motion, i y/ — 
 
 or time of defcribing AF : AF or v^2R x~AD : : 
 
 / : \/2Rs iz the velocity of the body, or fpace 
 defcribed in time /. 
 
 Cor. I . The velocity of the revolving hcdy^ is equal 
 to that which a falling body acquires in defcending 
 thro' half the radius AC, by the force at A uniform- 
 ly continued. 
 
 For v^J (hight) : is (the velocity) : : v/|R (the 
 
 hight) : v^zRj, the velocity acquired by falling 
 thro' i R. 
 
 Cor. 2. Hence^ if a body revolves uniformly in a 
 tircle^ by means of a given centripetal force ; the 
 arch which it defcribes in any time, is a mean propor- 
 tional between the diameter of the circle, and the 
 fpace which the body would defcend thro" in the fame 
 time, and with the fame given force. 
 
 For 2R (diameter) : v/2Rj : : v/2Rj : s\ where 
 
 y/2Rj is the arch defcribed, and s the fpace defcend- 
 ed thro', in the time /. 
 
 2. Cor. 1. If a body revolves in any curve AFQ, 
 about the center of force s •, and if AC or R be the 
 radius of curvature in any point A\ s zi. fpace defcend- 
 ed by the force directed to C. Then the velocity in A 
 
 w/7/ be y/2R.r. 
 
 For this is the velocity in the circle •, and there- 
 fore in the curve, which coincides with it. 
 
 PROP.
 
 Sea. I. CENTRIPETAL FORCES. 5 
 
 Fig. 
 PROP. III. 
 
 If fever al bodies revolve in circles round the fanie i« 
 or different centers-^ the periodic times will he as the 
 fquare roots of the radii direElly^ and the fquare roots 
 of the centripetal forces reciprocally. 
 
 Let F rz centripetal force at A tending to the 
 center C of the circle. 
 V — velocity of the body. 
 R zz radius AC of the circle, 
 P ~ periodic time. 
 
 2R 
 
 Then (Prop. II.) P =: tt / ^Z But s is as the 
 
 2R 
 
 force F that generates it j whence P — tt / ^/'t^' 
 
 and fince 2, tt and / are given quantities, therefore 
 
 Cor. 1 . 1'he periodic times are as the radii dire5ily, 
 and the velocities reciprocally. 
 
 For (Prop. II.j V =: ^/2Rs - V/2RF, and V* 
 
 2R 
 — 2RF, and P = TT / v^X' ^^^ ^^ —-rrWx 
 
 2R 
 
 -p » therefore P' V' z= ttW^ X 4R% and P^ = 
 
 TtV^ X 4R' TT/ X 2R R 
 
 yl 5 andPiz y OCy- 
 
 Cor. 2. The periodic times are as the velocities di- 
 rectly-, and the centripetal forces reciprocally. 
 
 VV R 
 
 For V* zz iKs ■=! 2RF ; and R z: — p-j and y 
 
 V V R V 
 
 rz^-pOC-p- But(Cor. i.)P OCy OCy 
 
 B 3 Cor,
 
 6 CENTRIPETAL FORCES. 
 
 •pig. Cor. 3. If the periodic times are equal \ the velo- 
 i . cities, and alfo the centripetal forces, will be as the 
 radii. 
 
 For if P be given; then t;? and yj and p are 
 
 all given ratios. 
 
 Cor. 4. If the periodic times are as the fqiiare 
 roots of the radii •, the velocities will he as the fquare 
 roots of the radii, and the centripetal forces equal. 
 
 For (Prop. III. and Cor. 1.) putting \/R fo^ P> 
 
 we have ^/R CC s/-^ OCv^-Therefore i OC "Tp OC 
 
 v/R ... 
 
 -trr- , and ^R oc V, and ^/F is a given quantity. 
 
 Cor. 5. If the periodic times are as the radii ; the 
 ^velocities will he equal, and the centripetal forces re- 
 ciprocally as the radii. 
 
 R R 
 
 For putting R for P, we have R OC s/^ OC y > 
 
 I I 
 
 whence y/R OC — 7^, and i OC Tf; that is, R 
 
 OC -F' or the centripetal force is reciprocally as 
 xht radius ; and V is a given quantity. 
 
 Cor 6. If the periodic times are in the fefquiplicate 
 ratio of the radii •■, the velocities will he reciprocally as 
 the Iquare roots of the radii, and the centripetal forces 
 redprocally as the fquares of the radii. 
 
 R R 
 
 Put R'forP, then r' OCy/p OCy-,andR oC 
 
 I I I 
 
 -"Tp or RR 0Cp> and v/R OCy * 
 
 Cor. 7. If the periodic times he as the nth power 
 of the radius ; then the velocities will he reciprocally as 
 the n — i'^ power of the radii, and the centripetal 
 
 forces
 
 Sea. I. CENTRIPETAL FORCES. 7 
 
 forces reciprocally as the in —' V^ -power of the Fig; 
 radii. i , 
 
 R R 
 Put R" for P, then R"" OC %/ p OC y * Whence 
 
 P R O P. IV. 
 
 If fever al bodies revolve in circles round the fame i. 
 cr different centers j the 'velocities are as the radii di- 
 re^Ifyy ajid periodic times reciprocally. 
 
 For putting the fame letters as in Prop. III. we 
 
 have (by Prop. II.) V - v/IrJ z= ^2K¥ ; and P 
 
 V 
 
 OC-p(byCor.2.Pr.IlI.),andPF OCV, andF OC 
 
 -• Whence V zz ^iR¥ - J iK X X, and V* 
 
 2RV 2R R 
 
 = —^' and V 1= -pT OCp- 
 
 Cor. I. 'The velocities are. as the periodical times j 
 and the centripetal forces. 
 For we had PF oC V. 
 
 Cor. 2. The fqtiares of the velocities are as the ra- 
 
 dii and the centripetal forces. 
 
 For V - v/IrF. 
 
 Cor. 3. If the velocities are equal; the periodic 
 times are as the radii, and the radii reciprocally as 
 the centripetal forces. 
 
 For if V be given, its equal p is a given ratio -, 
 
 and v/RF is given, whence R oCp * 
 
 B 4 Cor,
 
 8 CENTRIPETAL FORCES. 
 
 Fiff. Cor. 4. If the velocities he as the radii, the perio- 
 I . ' die times will he the fame -, and the centripetal forces 
 
 as the radii. 
 
 R I 
 
 For then V or R OC p , and i OC p * Alfo R = 
 
 ^2RF, whence R OCF. 
 
 Cor. 5. If the velocities he reciprocally as the radii-, 
 the centripetal forces are reciprocally as the cubes of 
 the radii i and the periodic times as the fquares of the 
 yadii. 
 
 For put ^ for V, then (Cor. 2.) ^ = \/2RF, 
 
 ^ = 2RF, whence F OC ^' Alfo ^ 0<^p' and 
 
 p ocRR- 
 
 PROP. V. 
 
 If fcjeral bodies revolve in circles about the fame 
 or different centers •, the centripetal forces are as the 
 radii dire^ly, and the fqiiares rf the periodic times 
 reciprocally. 
 
 Put the fame letters as in Prop. III. Then (Prop. 
 
 2R 2R 
 
 II.) V — ir t v/~ 1= TT / v/ f"? and PP = ttt/^ 
 
 2R , ^ lirnttK 
 
 X -^j and PPF 1= imrtt'K ; whence F zz — piT"" 
 
 R^ 
 
 OC pp- 
 
 Cor. I . T^he centripetal forces are as the velocities 
 dire^ly-, and the periodic times reciprocally. 
 
 R , R V 
 
 For (Prop. IV.) V OCp^andF OCpp OCp- 
 
 Cor. 2. The centripetal forces, are as the fquares 
 of the velocities dire^ly, and the radii reciprocally. 
 
 For 
 
 I.
 
 Sea. I. CENTRIPETAL FORCES. 9 
 
 For (Cor. i.)F OCp, andFP OCV. But (Prop, if 
 
 III. Cor. I.) P 0Cy5 therefore FP OC-y"' there- 
 
 , FR ^^ , ^ VV 
 fore -y- OCV, and F OC "h— 
 
 Cor. 3. If the centripetal forces are equal \, the 've- 
 locities are as the periodic times \ and the radii as the 
 fquares of the periodic times, or as the fqiiares of the 
 velocities. 
 
 Cor. 4. If the centripetal forces be as the radii, 
 the periodic times will be equal. 
 
 R F I F 
 
 For F ocpp* and ^ OCpp, and if ^ be a 
 
 given ratio, pp will be given, as alfo P. 
 
 Cor. 5. If the centripetal forces be reciprocally as 
 the fquares of the difiances \ the fquares of the peri- 
 odical times will he as the cubes of the difiances ; and 
 the velocities reciprocally as the fquare roots of the 
 difiances. 
 
 I I R 
 
 For writing |>^ for F, then ^ ^pp' ^^^ 
 
 R' . . ^ ^ I VV I 
 
 pT a given quantity. And ^ ^IT' and -^ OC 
 
 VV, orv/ j^ OCV. 
 
 P R O P. VI. 
 
 If fever al bodies revolve in circles., about the fame '• 
 cr different centers \ the radii are direfdy as the centri- 
 petal forces, and the fquares of the periodic times. 
 
 For (Prop. II.) putting the fame letters as be- 
 
 r o 2R 2R 
 
 fore, P — TT / ^~ ~ TT t v/-p^' andPP — -mrtt 
 
 2R 
 
 X -p, and PPF =: i-rrirttV. OC R. 
 
 Cor.
 
 to CENTRIPETAL FORCES. 
 
 Fig. Cor. I . I^he radii are direElly as the velocities and 
 I. periodic times. 
 
 For (Prop. IV. Cor. i.) PF OC V, but PPF OC 
 R; therefore PV ocR. 
 
 Cor. 2. The radii are as the fquares of the veloci- 
 ties direSlly-, and the centripetal forces reciprocally. 
 
 V 
 For (Prop. III. Cor. 2.) P oC y' but (Cor. i.) 
 
 VV 
 R OC PV J therefore R OC -jft * 
 
 Cor. 3, If the radii are equal -^ the centripetal forces 
 
 are as the fquares of the velocities, and reciprocally 
 
 as the fquares of the periodic times. And the veloci- 
 
 ties reciprocallly as the periodic times. 
 
 VV 
 For if R be given, -p- j and PPF, and P V,are given 
 
 quantities, and F OCVV, orF OCpp'andV OCp* 
 
 Scholium. 
 The converfe of all thefe propofitions and corol- 
 laries are equally true. And what is demonftrated 
 of centripetal forces, is equally true of centrifugal 
 forces, they being equal and contrary. 
 
 PROP. VII. 
 
 I, The quantities of matter in all attracting bodies, 
 having others revolving about them in circles -, are as 
 the cubes of the diftances dire5tly, and the fquares of 
 the periodical times reciprocally. 
 
 Let M be the quantity of matter in any central 
 attradling body. Then fince it appears, from all 
 aftronomical obfervations, that the fquares of the 
 periodical times are as the cubes of the diftances, 
 of the planets, and fatellites from their refpeftive 
 
 centers.
 
 Sea. I; CENTRIPETAL FORCES. n 
 
 centers. Therefore (Cor. 6. Prop. 111.) the centri- Fig. 
 petal forces will be reciprocally as the fquares of the i, 
 
 diflancesi that is, F OC^^* And (Prop. I.) 
 
 the attradlive force at a given diftance, is as the 
 body M, therefore the abfolute force of the body 
 
 M is as p^' And (Prop. V.) fmce F OCwp, put 
 
 M . n . ri- J 1 M ^ R ■ 
 
 — inftead of F, and we have ^^ CC p^-> and 
 
 R' 
 
 M ocp- 
 
 Cor. I. He'dce infiead of F in any cf tl>e foregoing 
 
 propofitio'/is and their corollaries^ one may fubftitute 
 
 M 
 
 ■jTi^j "dchich is the force that the at tracing body in Cy 
 
 exerts at A. 
 
 Cor. 2. The attractive force of any hody^ is as 
 the quantity of matter dire^ly^ and the fquare of the 
 difiance reciprocally, 
 
 PROP. VIII. 
 
 If the centripetal force he as the difiance from the o, 
 center C. A body let fallfroin any point A, will fall 
 to the center in the fame tifne, that a body revolving 
 in the circular orbit A LEA, at the difiance CA, 
 would defcrihe the quadrant AGL. 
 
 The truth of this is very readily fhewn by fluxions; 
 thus, putAC— r, API— ;f, f — time of defcrib- 
 jng AH, V zz the velocity at H. F — force at H, 
 
 which is as CH or r — x. Then (Mechan. Cor. 2. 
 Prop. V.) the velocity generated is as the force and 
 
 time •, that is, v OCF t. Alfo (Median. Prop. III. 
 Cor. I . ) the time is as the fpace divided by the veloci- 
 ty,
 
 12 CENTRIPETAL FORCES. 
 
 Fig. • X . F;^ r x'Xx 
 
 Q tv ; that is, / OC — i therefore vQC — CC a 
 
 'VV 
 
 and -ui; OC r — ;^ X ^5 and the fluent is — OC rx— 
 
 XX , y 
 
 — , OYVV OCirx — AT^.andi; oCv 2rx — xx or 
 2 
 
 HG -, that is, the velocity at H is as the ordinate 
 HG of the circle. 
 
 Now it is evident, that in the time the revolv- 
 ing body defcribes the infinitely fmall arch AF, the 
 falling body will defcend thro' the verfed fine AD, 
 and would defcribe twice AD in the fame time, 
 with the velocity in D. Therefore we fliall have, 
 velocity at F : velocity at D : : AF or FD : 2 AD, 
 and velocity at D : velocity at H : : AF or FD : GH, 
 
 therefore, 
 velocity at F or G : velocity at H : : AF* : 2 AD X 
 
 AF" 
 GH : : -tj^ : GH : : CA or CG : GH. But 
 2 AD 
 
 drawing an ordinate infinitely near GH-, by the 
 
 nature of the circle, it will be, as GC : GH : : fo 
 
 the increment of the curve AG : to the increment 
 
 of the axis AH. And therefore, vel. at G : veL 
 
 at H : : as the increment of AG : to the increment 
 
 of AH. Therefore fince the velocities are as the 
 
 fpaces defcribed, the times of defcription will be 
 
 equal ; and the feveral parts of the arch AGL are 
 
 defcribed in the fame times as the correfpondent 
 
 parts of the radius AHC. And by compofition, 
 
 the arch AG and abfciifa AH, as alfo the quadrant 
 
 AL and radius AC, are defcribed in equal times. 
 
 Cor. I . '^he velocity of the defcendit?g body at any 
 flace H, is as the fine GH. 
 
 Cor. 2. j4nd the time of defcending thro'' any verfed 
 fine AH, is as the correfponding arch AG. 
 
 Cor.
 
 Sea. I. CENTRIPETAL FORCES. 13 
 
 Cor. 3. All the times of falling from any altitudes Fig. 
 whatever, to the center C, will he equal. 3. 
 
 For thefe times are -l the periodic times ; and 
 (Prop. V. Cor, 4.) thefe periodic times are all equal. 
 
 Cor. 4. In the time of one revolution, the falling 
 hody will have moved thro' C to E, and hack again 
 thro^ C to Ay meeting the revolving hody again at A. 
 
 Cor. 5. The velocity of the falling hody at the cen- 
 ter C, is equal to the velocity of the revolving hody. 
 
 For the velocities are as the lines GH and GC ; 
 and thefe are equal, when G comes to L. 
 
 PROP. IX. 
 
 If a pendulum AB he fiifpended at A, and he made 4» 
 to revolve by a conical motion, and defcribe the circle 
 BEDH parallel to the horizon. 
 
 Put Tc — 3.1416; p =1 164-^ feet, the fpace de* 
 fcended hy gravity in the time t. 
 
 2 AC 
 
 Then the periodical time of B will he -kI y/ —r— * 
 
 For (Mechan. Prop. VIII.) if the axis AC re- 
 prefents the weight of the body, AB will be the 
 force ftretching the ftring, and BC the force tend- 
 ing to the center C. Alfo (Mechan. Prop. VI.) if 
 the time is given, the fpace defcribed will be as the 
 
 BC 
 force i whence AC : BC : : ^ : : xp P — ^^e fpace 
 
 defcended towards C, by the force BC, in the time 
 /. This is the fpace s in Prop. II. Therefore in- 
 ftead of s put its value in the periodical time, and 
 (by Prop. II.) we fhall have the periodical time of the 
 
 2R /TT AC~ 
 
 pendulum zztt t -y "t zz. t: t ^ 2BC X gpTrr 
 
 ,2AC 
 
 = TT t v/-p- • 
 
 Cor.
 
 14 CENTRIPETAL FORCES. 
 
 Fig. Cor. I . In all pendulums, the periodic times are as 
 4. the fquare roots of the bights of the cones, AC. 
 For TT, /, and p are given quantities. 
 
 Cor. 2. If the bights of the cones be the fame, the 
 periodic times will be the fame, whatever be the radius 
 of the hafe BC. 
 
 Cor. 3. Hhe femiperiodic time of revolution, is 
 
 equal to the time of of dilation of a pendulum, ivhofe 
 
 length is AC, the bight of the cone. 
 
 AC 
 For by the laws of falling bodies, / ^y — zz 
 
 time of falling thro' ^ AC -, and therefore (Me- 
 chan. Prop. XXIV.) i : tt : : t k/ — : tt / v/ — 
 
 JT / ^ ip ^ 1p 
 
 z=. {-^t \/—--% the time of vibration, which is 
 half the periodical time. 
 
 Cor. 4. 7'he fpace defcendcd by a falling body, in 
 
 the time of one revolution, will be tttt X 2 AC. 
 
 2 AC 
 For tt (time) : p (hight) : : trirtt x —- (per. 
 
 time) : ttttX 2 AC — hight defcended in that time. 
 
 Cor. 5. ^he periodic time, or time of one revolu- 
 
 tioit, is equal to -\/i X time of falling thro'' AC. 
 
 AC 
 For the time of falling thro' AC is / n/"' 
 
 Cor. 6. T^he weight of the pendulum is to the cen- 
 trifugal force; as the hight of the cone AC, to the 
 radius of the bafe CB. And therefore when the hight 
 CA is equal to the radius CB ; the centripetal or 
 centrifugal force is equal to the gravity. 
 
 PROP.
 
 Sea. I. CENTRIPETAL FORCES. 15 
 
 P R O P. X. ^ 
 
 Suppofe a fyjiem of bodies A, B, C, to revohe 5. 
 with a conical motion about the axis TR perp, to the 
 horizon, fo as to keep the fame fide always towards 
 the axis of revolution, and the fame pojition among 
 themf elves. 
 
 I0 find the periodical time of the whole fyfiem* 
 
 I . Let A, B, C be all fitiiated in one plane 
 pafling thro' TR. From A, B, C let fall the 
 perpendiculars Ka, B^, Cf, upon the axis TR. 
 And let A, B, C reprefent the quantities of matter 
 in the bodies A, B, C. Alfo put h zz i6^V feet, 
 the hight a body falls in the time / by gravity •, x 
 — 3. 141 6 ; P zz the periodic time of the fyftem. 
 
 Uy the refolution of forces, Ta (gravity) : Atf 
 
 ^a 
 (force in diredtion Ka) w h \t^ h — fpace de- 
 
 fcended by A towards a in the time i, which is as 
 the velocity generated by the force Ka. There- 
 fore rjT hK zz motion generated in A in diredlion 
 
 Aa. And the force in direftion A^ to move the 
 fyllem towards TR, by the power of the lever 
 
 r^ . • A^ , , ^ , , . _, . . . 
 
 TA, IS ;p- i&A X Ta or Aa X hA. This is the 
 
 centripetal force of the fyilem, arifmg from the gra- 
 vity of A. In like nianner the centripetal forces 
 arifmg from B and C, will be Bb X i?B and Qc X hC 
 By the laws of uniform motion, P : 2-,r X Aa : : 
 27rt X Aa 
 t : p ~ arch defcribed by A in the time 
 
 ^TTTTtt X Aa" i^TTtt X Aa 
 
 i' And pp X '^TAa^^ PP ~ diftance it 
 
 is drawn from the tangent in that time, or as the 
 
 , . , ^ , r 2 7r~// X Aa- 
 
 velocity generated ; and therefore - — prj X A
 
 i6 CENTRIPETAL FORCES. 
 
 Fig, =. motion of A tending from the center a, by the 
 5. revolution of the fyflem. And the force in direc- 
 tion aA, to move the fyflem from TR, by the 
 
 27rTr// X Aa 
 
 power of the lever TA, will be pp A X 
 
 Ta. And this is the centrifugal force of the fyf- 
 tem arifing from the revolution of A. And in like 
 manner the centrifugal forces arifing from B and 
 
 C, will be pp B X T^, and pp C 
 
 X Tc. 
 
 But becaufe the whole fyftem always keeps at 
 the fame diftance from the axis TF, in its revolu- 
 tion; therefore the fum of all the centripetal forces 
 muft be equal to the fum of all the centrifugal 
 forces. Whence A^ X ^ A zz B^ X y^B -h C<: X hC — 
 
 -pp- X into A« X T^ X A -f- B^ X T^ X B + Cc 
 
 2 
 
 X Tf X C. And confequently P = ^ / ^-r X 
 
 A^? X T^ X A -^ B^ X Ti' X B -^CcXTcxC 
 Aa X A -I- B^ X B 4- C^ X C 
 2. If the bodies are not all in one plane, let N 
 be the center of gravity of the bodies A, B, C. 
 And thro' N draw the plane TNR; and from all 
 the bodies, let fall perpendiculars upon that plane. 
 Then the periodic time will be the fame as if all 
 the bodies were placed in thefe points where the 
 perpendiculars cut the plane. For if jjj be one ot 
 the bodies, and mC perp. to the plane. Then the 
 centripetal and centrifugal forces of m in dire6tion 
 
 27r7r// X Tc 
 
 cm, will be cm X hm and pp ?n X mc. But 
 
 the force cm is divided into the two forces cC, Cm. 
 And all the forces Cw deftroy one another, be- 
 caufe the plane, TNc, pafles thro' their center of 
 gravity. Therefore the plane is only adcd on by 
 
 the
 
 Sea. I. CENTRIPETAL FORCES. 17 
 
 the remaining force <:C. So that the centripetal Fig. 
 and centrifugal forces will be the fame as before, 5. 
 when the body was placed in C ; and the periodic 
 time is the fame. 
 
 Cor. I . If N;z be drawn from the center of gra- 
 vity perp. to TFj then the periodic time of the fyf- 
 
 2 
 Urn, P iz TT / ^-j X 
 
 T^ X A^ X A 4- T^ X B^ X B 4- Tr X C<: X C 
 
 N« X A + B + C 
 For (Mechan. Prop. X XXV.) A^ X A + B^ X 
 
 B + Cc X C zz N;? X A + B + C. 
 
 Cor. 2. 'The length of a Jimple pendulum, making 
 two vibrations, or an exceeding fmall conical motion^ ■ 
 in the fame periodic time, will be 
 T^X A^X A + T/^X B^X B + TrXCfXC 
 
 N;^ X A + B + C 
 For let TO be the hight of the cone defcribed 
 
 iTTTZtt 
 
 by the pendulum ; then (Prop. IX.) PP r= — t~ X 
 
 TO ; therefore TO ■=. 
 
 TaXAaxA + T/^ X B/* X B + T^ X Cr X C 
 
 '^n X A + B~+ C 
 
 Cor. 3. If TO be the length of an ifocrcnal pen- 
 dulum, then O is the center of gravity of all the pe- 
 ripheries defcribed by A, B, C ; each rauhiplied by 
 the body ; whether A, B, Q be the places of the bo- 
 dies, or the points of proje.. ion upon the plane TNR. 
 
 For if Aa X A, M X 6, Qc X C be taken for 
 bodies, their center of gravity will be diftant from 
 T, the length 
 T^ X A^X A -f- Tb X B^- X B -u T^ X Cr X C 
 
 Arz X A 4- b^ X B -t- ^^ A C ' ^^^^ 
 
 C chan.
 
 i8 CENTRIPETAL FORCES. 
 
 Fig. chan. Prop. XXXV.) which is equal to TO by 
 5. Cor. 2. and the peripheries are as the radii, A^, B^, 
 Cc, 
 
 Cor. 4. If any of the bodies he on the contrary fide 
 of the axis TR, or above the 'point of fufpenfton T j 
 
 that difiance rnuft be negative. 
 
 Cor. 5. If any line or plane figure be placed in the 
 plane TNR •, then the point O, "duhich gives the length 
 of the pendulum^ will be the center of gravity^ of 
 the furface or folid, defcribed in its revolution. 
 
 Scholium. 
 
 The point O which gives the length of the ifo- 
 cronal pendulum •, is called the center of turbina- 
 tio?i or revolution. And the plane TNR paffing 
 thro' the center of gravity, the turbinati7ig plane. 
 
 SECT.
 
 [ '9] 
 
 SECT. II. 
 
 The 77iotion of bodies in all forts of 
 Curve Lines. 
 
 PROP. XI. . 
 
 cr'HE areas ^ which a r evolving body defcrihes hypi^ 
 radii drawn to a fixed center of force^ are propor- 5^ 
 tional to the times of defcription ; and are all in the 
 fame immoveable plane. 
 
 Let S be the center of force -, and let the time 
 be divided into very fmall equal parts. In the firft 
 part of time let the body defcribe the line AB 5, 
 then if nothing hindered, it would defcribe BK zz 
 AB, in the fecond part of time -, and then the area 
 ASB — BSK. But in the point B let the centri- 
 petal force aft by a fingle but ftrong impulfe, and 
 caufe the body to defcribe the line BC. Draw KC 
 parallel to SB, and compleat the parallelogram 
 BKCr, then the triangle SBC = SBK, being be- 
 tween the fame parallels ; therefore SBC — SBA, 
 and in the fame plane. Alio the body moving uni- 
 formly, would in another part of time defcribe Cm 
 •=z CB ; but at C, at the end of the fecond part of 
 time, let it be a6led on, by another impulfe and 
 carried along the line CD •, draw mT) parallel to 
 CS, and D will be the place of the body after the 
 third part of time •, and the triangle SCD — SQm 
 =z SCB, and all in the fame plane. After the 
 fame manner let the force adt fuccefiively at D, 
 E, F, &c. And making Tin — DC, and E<? z= 
 ED, &c. and compleating the parallelograms as 
 C 2 before;
 
 20 CENTRIPETAL FORCES. 
 
 Fig before -, the triangle CSm ~ CSD zz DSn = DSE 
 6. — ESo rr ESF, &c. and all in the fame im- 
 moveable plane. Therefore in equal times equal 
 areas are defcribed ; and by compounding, the fum 
 of all the areas is as the time of defcription. Now 
 let the number of triangles be increafed, and their 
 breadth diminifhed ad infinitum ; and the centri- 
 petal force will aft continually, and the figure 
 ABCDEF, &c. will become a curve ; and the areas 
 will be proportional to the times of defcription. 
 
 Cor. I. If a body defcrihes areas proportional to 
 the times^ about any point -, it is urged towards that 
 point by the centripetal force. 
 
 For a body cannot defcribe areas proportional to 
 the times, about two di^erent points or centers, in 
 the fame plane. 
 
 Cor. 2. I'he velocity of a body revolving in a 
 curve., is reciprocally as the perpendicular to the tan- 
 gent^ in that point of the curve. 
 
 For the area of any of thefe little triangles being 
 given ; the bafe (which reprefents the velocity) is 
 reciprocally as the perpendicular. 
 
 7* Cor. 3. 'The angular velocity at the center of 
 force., is reciprocally as the fqiiare of its difiance from 
 that center. 
 
 For if the fmall triangles CSD and SBA be 
 equal, they are defcribed in equal times. The 
 SC X CQ , SB X BP 
 
 area CSD — » and area SBA — — > 
 
 2 2 
 
 therefore SC X CQ^zz SB x BP. But the angle 
 
 CSD : angle ASB : : CQ^: cci : : SC X CQ^: SC X 
 
 c^ : : SB X BP : SC X cq^ : : area SBA : area S^^ : : 
 
 SB' : Sc"- or SC\ 
 
 PROP.
 
 Sea. II. CENTRIPETAL FORCES. ^i 
 
 XT' 
 
 PROP. XII. gf 
 
 If a body revolving in any curve VIL, he urged by 
 a centripetal force tending towards the center S -, the 
 centripetal force in any point I of the curve will be as 
 
 P 
 
 —Ti where p zz. perpendicular S? on the tangent at 
 
 p'd 
 
 I, and d =z the difiance SI. 
 
 For take the point K inSnitely near I, and draw 
 the lines SI, SK; and the tangents IP, K/; and 
 the perpendiculars SP, S/. Alfo draw K»z, K» 
 parallel to SP, bl, and KN perp. to SI. 
 
 The triangles ISP, IKN. «K?w, are fimilar; as 
 alfo lK.m, l?q. Therefore Iq or IP : IK : : q? : 
 Km, And PS : IP : : K/« : mn. And IN : IK : : 
 mn : nY^. And multiplying the terms of thefe three 
 proportions, IP X PS X IN : IK X IP X IK : : 
 a? X K»2 X mn : Kw X mn X nK. That is, PS X 
 
 yq X IK^ 
 IN : IK^ : : ?P : «K n: pg ^ j^^ - But (Mechan. 
 
 Prop. VI.) the fpace ;?K, thro' which the body is 
 drawn from the tangent, is as the force and fquare 
 of the time-, that is (Prop. XI.) as the force and 
 fquare of the area ISK, or as the force X SP X 
 KN\ or becaufe SI X KN = twice the triangle 
 ISK r: IK X SP i therefore nYi. is as the force X 
 IK'' X PS'. Therefore the force at I is as 
 
 nK Vq X I K^ P^ _ 
 
 IK^PS^ -PSXINXIK^XPS^ ~PSj XIN - 
 
 A. 
 f^'d 
 
 »K " 
 Cor. I. The centripetal force at I is as -^rr—^^^^ 
 
 nK. 
 
 ^^"^•^sFxTk^* C 3 Cor.
 
 22 CENTRIPETAL FORCES. 
 
 Fig. Cor, 2. Hence the radius of curvature in I, is zz 
 87 SIX IN 
 
 IK* 
 
 For that radius =: g- = (by the fimilar trian- 
 
 IK X IP 
 
 gles IKw, Iq?) — p zi (by the fimilar trian- 
 
 SI X IN 
 
 gles IPS, INK) — p— • 
 
 PROP. XIII. Proh, 
 
 To find the law of the centripetal force^ requifite 
 to make a body move in a given curve line. 
 
 Let the diftance SI zz d, the perpendicular SP 
 (upon the tangent at I) zz p -, then from the na- 
 ture of the curve, find the value of p in terms of 
 d^ and fubftitute it and its fluxion, in the quantity 
 
 fd 
 
 Or find the value of oi- .> t^vtx or 
 
 SP X KN* "' SP^ X IK' 
 Any of thefe will give the law of centripetal force, 
 by the lalt Prop. 
 
 Ex. I. 
 
 If a body revolves in the circumference of a cir-^ 
 cle \ to find the force directed to a given point S. 
 
 Draw SI to the body at I, SP perp to the tan- 
 gent PI, SG perp. to the radius CI. Then SP zz 
 GI •, becaufe SGIP is a parallelogram. Put SI zz 
 d, SP ZI p, SC z: ^, CI ZI r, CD iz x, ID be- 
 ing perp. to SD. Then in the obtufe angle SCI, 
 SP ZI SC^- + CP -h ^SCD, or dd - aa + rr '\^ 
 dd — aa — rr 
 
 lax'y whence x zz The triangles 
 
 2a ^ 
 
 SCO and CID are fimilar, whence CI (;') : CD
 
 Sea. II. CENTRIPETAL FORCES. 23 
 
 / \ en r ^ rn ^^ ^'^ — aa—rr ^ Fig. 
 
 (*) : : SC {a) : CG = - zz ~ j and ° 
 
 dd — aa — •" rr dd + rr — aa 
 p zz r '\- — J and 
 
 $ zz—' Therefore the force f -A ) is as — r = 
 
 d dX 8r^ 
 
 rr, — ^, 1, , 3 s that is, the force is as 
 
 ^P r X dd -\- rr — aa 
 
 d 
 
 dd -^ rr — aa 
 
 And if £1 zz r, the force is as js' 
 
 Ex. 2. 
 
 If a body revolves in an ellipjts ; 4o find the force 10. 
 tending to the center C. 
 
 Let -'- tranfverfe CV zz r, -i conjugate CD zz f, 
 draw CI zz d, and its femiconjugate CR zz b^ 
 Then by the properties of the ellipfis fCon. Sed: 
 B. I. Prop. XXXIV.) hb + dd zi rr -\- cc^ 
 whence b zz ^/rr + cc — dd -^ and (ib. Prop. 
 XXXVII.) b or v/rr -^ cc ^Id '. c : : r : p zz 
 
 ^1 J . crdd 
 
 V^T^—ZTTd •' -^ ^ = TTTT^ZTJ^ There. 
 
 /)* rrii/ rr + Cf — -~^^' ^ 
 
 fore -^ =z - ^r— ■ .3 X : zz ~ • 
 
 p'd rr + cc — dd' "■ c^r'd 
 
 Therefore the force is dire6lly as the diftance CI. 
 After the fame manner, the force tending to the 
 
 center of ah hyperbola, will be found — ;:;, which 
 
 is a centrifugal force, dlredly as the diftance. 
 Ex. 3. 
 Jf a body revolves in an ellipfis^ to find the law of ^m 
 centripetal foire, tending to the focus S^ 
 
 C 4 Let
 
 24 CENTRIPETAL FORCES. 
 
 Fig. Let the femitranfYerfe OV — r, the femiconju- 
 II. gate OD rz r, draw SI — d; and OI, and its 
 conjugate OK — i?. 
 
 Then (Con. Sed. B. I. Prop. XXXV .) ird — d d 
 - bb i and (ib. Prop. XXXVL) b or y/^rd—dd : 
 
 c : : a : p = r- j and p zz 
 
 vird — dd 
 
 c'd s/ird -^d — cdy. ird — d^~^i Xrd — dd _ 
 2rd — dd 
 
 cdXird—dd — cd X rd — d'd _crdd 
 
 2ra — dd^ '2-rd — di\\ 
 
 ^, r P c^dd X 2^r — difi ^^^ ^ 
 
 Therefore — : iz: - — : ' . — .TT :=.TZu 
 
 fd c''d-'dX2dr-ddli '^ ^'^^ 
 
 Therefore the centripetal force is s.s-rj'y or recipro-' 
 
 cally as the fquare of the diftance. 
 Ex. 4. 
 12. -!y^ ^ ^^4)' ^'^'^'ohes in the hyperbola VI ; to find the 
 laisj of cent' i petal force^ tending to the focus S. 
 
 Draw SI, and the tangent IT, and SP perp. 
 upon it. And let the femitranfverfe SO — r, fe- 
 miconjugate zz V, SI ~ ^, SP — p^ and b n: fe- 
 miconjugate to lO. 
 
 Then i^Con Seft. B. II. Prop. XXXI.) ird + 
 
 dd — bb, and b — ^ird + dd. And (ib. Prop. 
 
 XXXII.) b or y/Q.rd + dd : c : : d : p =z 
 
 cd 
 , — == ; whence p — 
 V^rd 4- dd 
 
 c'd X s/ird 4- dd — cd X zrd + dd '^ X rd -j- dd 
 2rd + dd 
 
 zz ^^ ^ ^^^ -^ ^^ "' ^^ ^ ^^ "^ ^ — — i^l— - 
 2rd -i- dd^'- 2?'^ + dd^^ 
 
 There-
 
 Sea. II. CENTRIPETAL FORCES. 25 
 
 ^, . p crdd xird -\- ddV- crd Fig. 
 
 Therefore A = r^-T- = — 5- = 12. 
 
 p^d 2rd^-dd)' d X c'^' ^'^^ 
 
 — -Ty* Therefore the centripetal force is as —r-, or 
 
 ^j that is, reciprocally as the fquare of the dif- 
 
 tance. 
 
 And in like manner the force towards the other 
 
 focus F, is — ^? or as -jj^ which is a centrifugal 
 force reciprocally as the fquare of the diftance. 
 
 Ex. 5. 
 
 If a hody revolves in the parabola VI ; to find the 1 3. 
 force tending to tht focus S. 
 
 Draw IS, and the tangent IT, and SP perp. to 
 it. And put SI ~, d^ SP — />, latus re6lum — r. 
 Then (Con. Sed. B. III. Prop. II. and Cor. 3. 
 
 Prop. XII.) pp zz ~ rd^ and 2pp — ^ rd; and p 
 
 _ ^'^ — ^^ A J -^ — ^^ 
 
 ^ Sp ^ ^\/rd p'd~~ ^^rdX^rdy/rdxd 
 
 8r 2- ^ , 
 
 •z=. T} — -Tf* Therefore the centripetal force 
 
 ^rrdd rdd ^ 
 
 is reciprocally as the fquare of the diflance CI. 
 
 Hence, in all the Conic Sedlions, the centripe- 
 tal force tending to the focus, is reciprocally as the 
 fquare of the diftance from the focus. 
 
 Ex. 6. 
 
 Let VI he the logarithmic fpiral^ to find the force 14, 
 tending to the center S. • '* 
 
 Draw the tangent IP, and SP perp. to it, let 
 51 == ^, SP — /> i then the ratio of ^ to p is al- 
 ways
 
 26 CENTRIPETAL FORCES. 
 
 ^^'ways given, fuppofe as m to /;. Then^ = -zA 
 
 15- 
 
 and p — —d. Conlequently —rzz — X "TTTr = 
 
 — ^ ; and the centripetal force is as ^r* or reci- 
 procally as the cube of the diftance. 
 
 PROP. XIV. 
 
 The 'velocity of a body moving in any curve QAO, 
 in any foint A j is to the velocity of a body moving 
 in a circle at the fame diftance ; as \^p'^ to V^p, 
 
 Putting d — dijlance SA, and p — SP the perpendi- 
 cular on the tangent at A. 
 
 Let AR be the radius of curvature •, from the 
 point a in the curve infinitely near A, draw am., 
 an parallel to AS, AR. Let C — velocity in the 
 curve, c zz velocity in the circle. By fimilar tri- 
 angle SP (p) : SA (d) : : an : am : : centripetal 
 force tending to R : centripetal force tending to 
 
 ^ CC cc 
 
 IJk^ ^^'°^' '-^ AR = AS- ^"' (P^°P- 
 
 XII. Cor. 2.) AR = ^ ; whence p:d::^: 
 
 '-::.. p dd 
 
 CC ' • 
 
 -2 : : CCp : ccd. And pd : dp :: CC : cc. 
 
 Cor. I. If r •=! half the tranfverfe axis of an el- 
 lipfis -y then the velocity of a body revolving round the 
 focuSy is to that in a circle at the fame dijlance ; a^ 
 
 \/xr — d : \/r. 
 
 ' ■For?' = — ^^ (See Ex. 3. Prop. XIII.), 
 ird- — dd\^
 
 l.l.jPa^. 2^.
 
 m 
 
 PI. I. /--A
 
 Sea. II. CENTRIPETAL FORCES. 27 
 
 zndp = ' , — . And the fquares of the ^^' 
 
 velocities in the curve, and in the circle, are as 
 
 cdd crddd , rd 
 
 • J — == and - , i or as i and —-7 t^j 
 
 s/ird-^dd 2rd-^dd)i 2rd-~dd 
 
 or as 2r — d to r. 
 
 Cor. 2. Suppofe as before^ the velocity of a body 
 revolving round the center of an ellipjis^ is to the ve- 
 locity in a circle at the fame dijtance ; as half the con- 
 jugate diameter to that difiance^ is to the diftance. 
 
 For p ~ — , and p = 
 
 vrr 4- cc — dd 
 
 crdd 
 
 Whence, the fquares of thefe ve* 
 
 rr -^ cc — dd^"^ 
 
 ... crd , crddd 
 
 locities are as and 
 
 V rr 4- cc — dd rr + cc —-aai * 
 
 dd 
 or as I to — ^^ ^j or as rr -\- cc — dd to dd, 
 
 or as bb to dd. See Ex. 2. Prop. XIII. 
 
 Cor. 3. 'J'he velocity in a parabola round the focus^ 
 is to the velocity in a circle at the fame diftance ; as 
 
 y/Z to I. 
 
 r'd 
 For p — ^ \/rd, and p — — — - (See Ex. 5. 
 
 4v/r^ 
 
 J>rop. XIII.) Whence the fquares of thefe veloci- 
 
 rdd 
 ties are zs ^d y/rd and — •y^-, or as ~rd to ^ rd-, 
 
 that is as 2 to I. 
 
 Cor. 4. The velocity of a body in the logarithmic 
 Jpiral in any pointy is the fame as the velocity of tt 
 ^ody at the fame dijiance in a circle* 
 
 For
 
 2S CENTRIPETAL FORCES. 
 
 ^^f For/. = ^^, and^ = £i (Ex. 6. Prop. XIIL) 
 
 n • 
 
 And the fqiiares of the velocities are as —dd and 
 i- til. 
 
 m 
 —dd, that is, equal. 
 
 PROP. XV. Prob. 
 
 1 6, To find the force which ailing in dire^ion of the 
 ordinate MP, fhall caufe the body to move in that curve. 
 
 Draw mp parallel and infinitely near MP, and 
 Mi parallel to AP. Then the force is as the fpace 
 mr^ thro' which it is drawn from the tangent, in a 
 given time. But ms is the fluxion and mr the fe- 
 cond fluxion of the ordinate PM. Therefore 
 making the fluxion of the time confl:ant j or which 
 is the fame thing, making the fluxion of the axis 
 confl:ant ; find the fecond fluxion of the ordinate, 
 which will be as the force. 
 
 Ex. I. 
 
 Let the curve be an ellipfis whofe equation isjy zz 
 
 c , 
 
 —VT.rx — ^x. Puttmg AP = y, PM —y, r = 
 
 femitranfverfe, c zz femiconjugate. Then y zz 
 
 JL V ^"^ ^'^ and V — — xs/irx — xx 
 
 ^ \/ irx — XX ^ 
 
 y 
 
 irx — i<x 
 
 "~- > X r — X y.rx — XX y. 2rx — xx "^ __ ^ ^ 
 
 ^rx — XX ^ 
 
 — 2rx + XX — r — X c — ^J" — cr^ 
 
 irx — xx\ i ^ irx — ~~xx\^ J* 
 
 That is, the force is as —^-^ or reciprocally as the 
 
 cube
 
 Sea. II. CENTRIPETAL FORCES. 29 
 
 cube of the ordinate. The fame is true of the Fig. 
 circle, which is one fort of ellipfis. 1 6. 
 
 Ex. 2. 
 Let the curve be a parabola, AP = x, PM = 
 y^ and rx ^=^ yy ', then rx zz 2yjf, and 2yy + 2yy 
 
 zz o ; therefore jyy n — y'y, and j iz — ~ z:-^ 
 
 rrxx — rr — i, 
 
 and the force as — r- or reci- 
 
 procally as the cube of the ordinate. 
 
 PROP. XVI. 
 
 If the law of centripetal force be reciprocally as 
 ihe fquare of the dijiance. The velocities of bodies 
 revolving in different ellipfes about one common cen- 
 ter -^ are directly as the fquare roots of the parame- 
 ters, and reciprocally as the perpendiculars to the tan- 
 gents at thefe points of their orbits. 
 
 Let J, D be the diftances in two ellipfes ; r, f, 
 /, p i and R, C, L, P, the femitranfverfe, femi- 
 conjugate, latus re6tLim, and perpendicular in the 
 two ellipfes. Then the fquares of the velocities in 
 two circles whofe radii are d, D, (by Prop. IV. 
 Cor. 2.) will be as ^ X force in d, and D X force 
 
 . ^ , . ^ D II 
 
 in D ; that is, as -Tj and tTtt or as t- and yc' 
 
 . Then (Prop. XIV. Cor. i.), 
 
 velocity in the ellipfis^: vel. in the circle i/: : </ 2r — d\t/r. 
 
 andvel.in thecircle</; vel. in theciiclcD:: v''-^ : i/fT* 
 
 And (Prop. XIV. Cor. i.) 
 
 vel. in the circle D : vel. in the ellipfis D :: v/R: V^zR — D. 
 Therefore vel. in the ellipfis d : vel. in the ellipfis 
 
 D::
 
 $d CENTRIPETAL FORCES. 
 
 Fig. ^ J ^r-^d X R . y 2R~^xT . . y2r—^ 
 
 ~D ' ^^ 
 
 But (Con. Sed. B. I. Prop. 36 )/> =z c \/^;ni-^> 
 
 2r — ^ 2r — d c 
 
 and ^ v^ — 2 — — ^•> ^'^^ ^y — 2 — ~T' ^"^ 
 
 2r — J c \/Tl , . cc 
 
 ^-ir ^J7r ^ -y-^b^^^"^^ 7 = ^ / by 
 
 2R--D 
 
 the Conic Sedions.) In like manner ^ — ^^ — ^ 
 
 — ■^— . Whence, yel. in the ellipfis d : vel. in the 
 
 v// v/L 
 
 elhpfisD: : ^^^-p"* 
 
 Cor. I . Hence the velocities in the two ellipfis, ere 
 2r — d 2R — D 
 
 ^s \/— ^— V and vZ-DlT"* 
 
 Cor. 2. Alfo the fquares of the areas defcribed in 
 the fame time^ are as the parameters. 
 
 For the areas are as the arches X perpendiculars, 
 or as the velocities X perpendiculars •, that is, as 
 
 \/l v/L 
 
 ~r^X p and -p- X P, or as <^l and ^/L. 
 
 Cor. 3. The velocity of a body in different parts of 
 
 its orbit is reciprocally as the perpendicular upon the 
 
 ir — d 
 tangent at that point j and therefore is as y/ — ^ — . 
 
 For the parameter is given. 
 
 Cor. 4. The velocity in a conic fe5tion at its great" 
 efi or leafi dijlance., is to the velocity in a circle at 
 
 the
 
 Sea. II. CENTRIPETAL FORCES. 31 
 
 the fame dijiance j as the fquare root of the parame- Fig* 
 ter^ to the fquare root of twice that dijiance. 
 
 For here ^ ~ D zr p — P, and L :=. 2D. There- 
 fore the velocity in the elHpfis, to the velocity in 
 
 the circle j as -5- : — g— : : y/l : v^2D. 
 
 Cor. 5. 'The 'velocity in an ellipfis at its mean dif- 
 tance, is the fame as in a circle at the fame dijiance. 
 
 For if d be the mean diftance, then p zz c. And 
 if D be the radius of the circle, then L zz 2D, 
 and P zr D. Whence, vel. in the ellipfis : to the 
 
 vel. in the circle : : — : ^^Ta— *• *• (becauferf z: ^ir) 
 
 *7V • 77iD • * ^^ • ^^* ^^^^ D = ^> there- 
 fore the velocities are equal. 
 
 Cor. 6. Both the real and apparent velocity round fj. 
 the focus F, Is great eft at A, the near eft vertex -, and 
 leaft at B, the remote vertex. 
 
 For the real velocity is reciprocally as the per- 
 pendicular, which is leaft at A and greateft at B. 
 And the apparent velocity at F is reciprocally as 
 the fquare of the diitance from F, which diftance 
 is leaft at A, and greateft at B, (Cor. 2. and 
 3. Prop. XL) 
 
 Cor. 7. The fame things fuppofed^ and PC, CK 23, 
 being femiccnj urates , the velocity in the curve ^ is to 
 the velocity towards the foats F; as CK to 
 
 v/ck^-^TtF. 
 
 For vel. in the curve : vci. towards F : : Pp : 
 
 pn : : FP : N? : : J : \/ dd — Jp. But pp = 
 
 crii . J. ird-^ dd — cc 
 
 -j> and da — Tf ^ — — j "' d. Whence 
 
 2r — d ^-^ 2r — a 
 
 vel in the cui"ve ; vel. towards F : : d :
 
 $2 CENTRIPETAL FORCES. 
 
 Fig. 2rd — dd — cc . ^ 
 
 23. s/ r^y. ^ d : : \/2rd — dd : \/2rd — dd — cc 
 
 s/cK' — CW. 
 
 Cor. 8. The afcending or defcending velocity is the 
 greatejl when FP is half the latus re£lu7n, or when 
 FP isprp. to AB. 
 
 For s/ird — dd : \/2rd — dd — cc : : vel. in 
 /v/2r — d\ 
 the curve y -j j : vel. towards F zi 
 
 2rd — dd — cc 
 v/ -ji ' ^^'^ making the fquare of this 
 
 . . 2rd — dd — cc 
 velocity a maximum, then -,-j zr ;», and 
 
 2rd — 2dd X dd — 2dd X 2rd — dd — cc = o ; 
 and rd — dd — 2rd -^ dd -\- cc zz o, and — rd 
 
 cc 
 
 -\- cc zzo. whence d =z — zz half the latus redlum. 
 
 r 
 
 Cor. 9. If FR, the dtjlance from the focus to the 
 
 curve be z= \/C A X CD •, then R is the place where 
 the angular motion about the focus F, is equal to the 
 mean ^notion. 
 
 For the area of a circle whofe radius FR is zz 
 
 V CA X CD is equal to the area of the ellipfis ; 
 and if we fuppofe them both defcribed in equal 
 times ; then the fmall equal parts at R will be de- 
 fcribed in equal times j and therefore the angular 
 velocities at F will be equal; and both equal to 
 the mean motion. The angular motion in the el- 
 lipfis from B to R will be flower ; and from R to 
 A fwifter, than the mean motion. 
 
 PROP.
 
 Se6l. II. CENTRIPETAL FORCES. 33 
 
 Fig. 
 
 PROP. XVIT. 
 
 If the centripetal forces he reciprocally as the 17* 
 fquares of the dijlances ; the periodic times in ellipfes^ 
 will he in the fefquiplicate ratio of the tranfverfe 
 <ixes AB ; or the fquares of the periodic times ^ will 
 he as the cuhes of the mean dijlances FD, from the 
 common center. 
 
 Put the fimbols as in the laft, and /, T, for the 
 periodical times. Then by the nature of the el- 
 lipfis f <r = 4. /r, and c zz y/ilr, and re zz r^^lr. 
 And for the fame reafon KC zz R^/^LR. Alfo 
 (Prop. XVI. Cor. 2.) the areas defcribed in the 
 fame time are as the fquare roots of the parame- 
 ters ; and therefore the whole areas of the ellipfes, 
 are as the periodical times multiplied by the fquare 
 roots of the parameters. But the whole areas are 
 alfo as the reftangles of the axes •, therefore the 
 reftangles of the axes are as the periodical times 
 multiplied by the fquare roots of the parameters ; 
 that is, re or r^^r : RC or Ry/^LR : : t^yl : 
 Tv/L. And fquaring, 4/r' : i^LR' ::///: TTL. 
 That is, r? : Rj : : // : TT. And / : T : : r^ : R^ : : 
 2r^ : 2R^. 
 
 Cor. I . 'The areas of the ellipfes are as the periodic 
 times multiplied by the fquare roots of the parameters. 
 
 Cor. 2. The periodic time in an ellipfis^ is the fame 
 as in a circle^ whofe diameter is equal to the tranfverfe 
 axis AB J or the radius equal to the mean diflance FD. 
 
 Cor. 3. The quantities of matter in central attraEl' 
 ing hodjesy that have others revolving ahout them in 
 ellipfes ; are as the cuhes of the mean diflauces^ divid- 
 ed by the fquares of the periodical times. 
 
 D For
 
 54 CENTRIPETAL FORCES. ^ 
 
 Flo-, For (Cor. 2.) the periodic times are the fame 
 I y, when the mean diftances are equal to the radii ; and 
 the reft follows from Prop. VII. 
 
 PROP. XVIII. 
 
 jO If the centriphal forces be dire5lly as the difiancesi 
 the periodic times of bodies moving in ellipfes round 
 the fame center^ will be all e^ual to one another* 
 
 Let AEL be an ellipfis, AGL a circle on the 
 fame axis AL, C the center of both. Draw the 
 tangent AD, and npY parallel to it, and D«, Bp 
 parallel to AC : AF being very fmall. Then D« 
 equal to B/> will be as the centripetal force ; and 
 therefore AD and AB, or A« and Ap will be de- 
 fcribed in the fame time, in the circle and ellipfis. 
 Confequently the areas defcribed in thefe equal 
 times will be A;;C and ApC. But thefe areas are 
 to one another as nY to PF, or as GC to EC ; that 
 is, as the area of the circle AGL to the area of 
 the ellipfis AEL. Therefore fince parts proportional 
 to the wholes are defcribed in equal times j the 
 wholes will be defcribed in equal times. And 
 therefore the periodic times, in the circle and el- 
 lipfis, are equal. 
 
 But (Prop. V. Cor. 4.) the periodic times in all 
 circles are equal, in this law of centripetal force ; 
 and therefore the periodic times in all ellipfes arc 
 equal. 
 
 Cor. "The 'velocity at any point I of an ellipfis, is 
 as the rectangle of the two axes AC, CE j divided 
 by the perpendicular CH, upon the tangent at I. 
 
 For the arch I X CH is as the area defcribed in 
 a fmall given part of time, and that is as the whole 
 area (becaufe the periodic times are equal) or as 
 AC X CE. And therefore the arch I or the velo- 
 • • AC X CE 
 city, is ^^ QYi ' PRO P.
 
 Sea. II. CENTRIPETAL FORCES. 35 
 
 Fig. 
 PROP. XIX. 18. 
 
 ^he denfitles of central attraSling bodies^ are reci- 
 procally as the cubes of the parallaxes of the bodies 
 revolving about them {as feen from thefe central bo- 
 dies)y and reciprocally as the fquares of the periodic 
 times. 
 
 For the denfity multiplied by the cube of the 
 diameter, is as the quantity of matter j that is (by 
 Prop. XVII. Cor. 3.) as the cube of the mean dif- 
 tance divided by the fquare of the periodical time 
 of the revolving body. And therefore the denfity 
 is as the cube of the diftance, divided by the cube 
 of the diameter, and by the fquare of the periodic 
 time. But the diameter divided by the diftance is 
 as the angle of the paralax ; therefore the denfity 
 is as I divided by the cube of the paralax,. and the 
 fquare of the periodic time. 
 
 PROP. XX. 
 
 If two bodies A, B, revolve about each other \ iq. 
 they "juill both of them revolve about their center of 
 gravity. 
 
 Let C be the center of gravity of the bodies A, 
 B, afting upon one another by any centripetal forces. 
 And let AZ be the diredtion of A's motion j draw 
 BM parallel to AZ, for the direftion of B. And 
 let AZ, BH be defcribed in a very fmall part of 
 time, fo that AZ may be to BH, as AC to BC j 
 and then C will be the center of gravity of Z and 
 H, becaufe the triangles ACZ and BCH are fimi- 
 lar. Whence AC : CB : : ZC : CH. But as the 
 bodies A and B attrad one another, the fpaces ha 
 and B^ they are drawn thro', will be reciprocally 
 
 D 2 as
 
 S6 CENTRIPETAL FORCES. 
 
 Fig. as the bodies, or direftly as the diftances from the 
 19. center of gravity, that is, Aa : Bl? : : AC : BC. 
 Compleat the parallelograms Ac and B^-, and the 
 bodies, inftead of being at Z and H, will be at c 
 and d. But fmce AC : BC : : A^ : B^. By divi- 
 fion AC : BC : : ^C : bC. But AC :BC : : AZ : 
 BH :: ac : bd. Whence aC : bC : : ac : bd. There- 
 fore the triangles cQa, and dCb are fimilar, whence 
 Cr : C^ : : ^f : Z'J : : AC : BC : : B : A. There- 
 fore C is ftill the center of gravity of the bodies 
 at c and d. 
 
 In like manner, producing B^ and Ar, till dg be 
 equal to B(i, and cq to Ac ; and if cf^ dh, be the 
 fpaces drawn thro' by their mutual attractions ; and 
 if the parallelograms ce^ di, be compleated. Then 
 it will be proved by the fame way of reafoning, that 
 C is the center of gravity of the bodies at q and g, 
 and alfo at e and f, where A defcribes the diagonals 
 Ac, ce, &c. and B the diagonals B^, di, &c. and 
 fo on ad infinitum. 
 
 If one of the bodies B is at reft whilft the other 
 moves along the line AL. Then the center of 
 gravity C will move uniformly along the line CO 
 parallel to AL. Therefore if the fpace the bodies 
 move in, be fuppofed to move in direftion CO, 
 with the velocity of the center of gravity j then 
 the center of gravity will be at reft in that fpace, 
 and the body B will move in direftion BH parallel 
 to CO or AZ i and then this cafe comes to the 
 fame as the former. Therefore the bodies will al- 
 ways move round the center of gravity, which is 
 either at reft, or moves uniformly in a right line. 
 
 If the bodies repel one another ; by a like rea- 
 foning it may be proved that they will conftantly 
 move round their center of gravity. 
 
 If the lines CA, Cc, Ce, &c. be equal ; and 
 CB, Cd, a, &CC. alfo equal. Then it is the cafe 
 of two bodies joined by a rod or a ftring j or of 
 
 one
 
 Sea. II. CENTRIPETAL FORCES. 'i^y 
 
 one body compofed of two parts. This body or Fio-. 
 bodies will always move round their common cen- 19. 
 ter of gravity. 
 
 Cor. I. 'The dire5fions of the bodies in oppojite points, 
 ef the orbits^ are always parallel to one another. 
 
 For fince AZ : Zc : : BH : H^; and AZ, Zc 
 parallel to BH, H^; therefore the <: ZAr zz <Z. 
 HBi, and B^ parallel to A^:. And for the fame 
 reafon di is parallel to ce^ &c. 
 
 Cor. 2 Two bodies^ a5ling upon one another by 
 any forces \ defcribe fimilar figures about their common 
 center of gravity. 
 
 . For the particles Ac, Bi of the curves are pa- 
 rallel to one another, and every where proportional 
 to the diftances of the bodies AC, BC. 
 
 Cor. 3. If the forces be directly as the diflances j 
 the bodies will defcribe concentrical ellipfes round the 
 center of gravity. 
 
 Cor. 4. If the forces be reciprocally as the fquares 
 of the diftances i the bodies will defcribe Jimilar el- 
 lipfes or fome conic fe5lions, about each other ^ whofe 
 center of gravity is in the focus of both. 
 
 PROP XXI. 
 
 If two bodies S, P attract each other with any 20. 
 forces^ and at the fame time revolve about their center 
 of gravity C. Then if either body P, with the fame 
 force,, defcrihes a fimilar curve about the other body S 
 at reft ; its periodical time^ will be to the periodical 
 time of either about the center of gravity ; as the 
 
 fquare root of the fum of the bodies fx/S + P), to 
 the fquare root of the fixed or central body fx/S). 
 
 Let PV be the orbit defcribed about C, and Vv 
 
 that defcribed about S. Draw the tangent Pr, tak^ 
 
 D 3 the
 
 3S CENTRIPETAL FORCES. 
 
 Fig. the arch PQ^ extremely fmall, and draw CQR ; 
 
 20. alfo draw Sqr parallel to CR, and then PQ and Pq 
 will be fimilar parts of the curves PV and P^'. 
 
 Now the times that the bodies are drawn from 
 the tangent thro' the fpaces QR, qr, with the fame 
 force, will be as the fquare roots of the fpaces QR, 
 qn that is (becaufe of the fimilar figures CPRQjind 
 
 SPrq) as \/CP to n/SP ; that is, (b y the nature 
 of the center of gravity) as \/S to v/S + P. Buc 
 the times wherein the bodies are drawn from the 
 tangent thro' RQ, rq^ are the times wherein the 
 fimilar arches PQ, Pq are defcribed -, and thefe 
 times are as the whole periodic times. Therefore 
 the periodic time in PV, is to the periodic time in 
 
 Vv i as v/S to v/S + pT 
 
 Cor. I. Tbe velocity in the orhit PV ahout C, is 
 ta the velocity in the orhit Pv about S ; as ^/S to 
 
 v/s"T'P^ 
 
 For the velocities are as the fpaces divided by 
 the times *, therefore, vel. in PV : vel. in Pi; : ; 
 PQ^ Pq _ C P SP S S-f P 
 
 v^"^ 'v/S~+P * ' -i/S • VsTP ' VS ' >/sTP 
 : : v/S : \/s + P. 
 
 Cor. 2. Bodies revolving round their common cen- 
 ter of gravity^ d:fcribe areas proportional to the times, 
 
 PROP. XXII. 
 
 20. If the forces he reciprocally as the fquares of th& 
 dijiances ; ayid if a body revolves ahout the center L 
 in the fame periodical time, that the bodies S, P, re- 
 volve about the center of gravity C. Then will SP : 
 
 LP : : VT^P : Vs. 
 
 Let PN be the orbit defcribed about L. Then 
 (Prop. XXI.) per. time in PQ : per. time in P^ : : 
 
 v/S;
 
 5ea. II. CENTRIPETAL FORCES. 39 
 
 V^S : v/sTP : : v/CP : v/5p7And (Prop. XVII.) Fig. 
 per. time in Fq : per. time in PN : : SFl : LP^" ; ^^' 
 iuppofing PQ, PN, fimilar arches. Therefore 
 per. time in PQ : per time in PN : : \/cp X SP^ : 
 ^SPxLP^ ; :\/cFxSF : v/EF. But the periodic 
 times are equal ; therefore \/CP x SP^ ::= v/lP*, 
 and LP^ =: CP X SP% a nd LP z= %/CP x SP\ 
 But LP : SP : : l/cFxSF : SP or ^SF ; : 
 (/CP : %/SP : : s/S : v^STp. 
 
 Cor. I. If the forces he reciprocally as the fquares 
 cf the diftances ; the tranfverfe axis of the elUpfis de~ 
 fcribed by P about the center of gravity C, is to the 
 tranfverfe axis defcribed by P about the other body S 
 at refl, in the fame periodical time ; as the cube root 
 of the fum of the bodies S + Pj /£> the cube root of the 
 fixed or central body S. 
 
 Cor. 2. If two bodies at tracing each other move 
 about their center of gravity. Their motions will be 
 the fame as if they did not attraEi one anotha\ but 
 were both attracted with the fame forces^ by another 
 lody placed in the center of gravity. 
 
 PROP. XXIII. Trob. 
 
 Suppofe the centripetal force to be directly as the 21.' 
 dijtance. To determine the orbit which a body will 
 defcribe^ that is proje^ed from a given place P, with 
 a given velocity^ in a given direSfion PT. 
 
 By Ex. 2. Prop. XIII. the body will move in 
 gn ellipfis, whofe center is C the center of force ; 
 and the line of direftion PT will be a tangent at 
 the point P. Draw CR perp. to PT. And let 
 thediftance CP zz d, CR — p^ femitranfverfe axis 
 D 4 CA
 
 40 CENTRIPETAL FORCES. 
 
 Fig. CA — R, femiconjugate axis CB — C. CG (the 
 21. femiconjugate to CP) — B. / iz fpace a body 
 would defcend at P, in a fecond, by the centripe- 
 tal force. V — the velocity at P, the body is pro- 
 jefted with, or the fpace it defcribes in a fecond. 
 
 Then s/zdf — velocity of a body revolving in a 
 
 circle at the diftance CP. 
 
 Then (Prop. XTV, Cor. 2.) v : v/2^/: : B : J, 
 
 and B\/2^/ — dv^ and 2BB<^ n: ddvvy whence 
 
 dvo d 
 
 BB — — /» and V> — v s/—a But (Con. Seft. B. 
 2/ ^ 2/ ^ 
 
 I, Prop. XXXI V.) RR + CC = BB + dd - 
 
 1)1)0, 
 
 —J + dd. And (ib. Prop. XXXVII.) CR z= Bp 
 
 d 
 zzpv^-r Therefore RR + CC + 2RC = 
 
 wd d 
 
 —r 4- dd -\- 2pv \/—r> and R -}- C z: 
 
 2/ ^ ^ 2/ 
 
 ywd d 
 
 -J -\- dd -{- 2pv ^—r = m. Alfo RR + CC 
 
 vvd , , d 
 
 ^ 2RC = -^ -\- dd^ 2pv ^—^ and R — C = 
 
 jwd d 
 
 s/ — 7 -{- dd — 2pv v/~r ~ «• Therefore R zz 
 2/ ^ ^ 2/ 
 
 ?« + « , ^ »z — ;/ 
 
 ' , and C — • 
 
 2 2 
 
 Then to find the pofition of the tranfverfe axis 
 AD. Let F, S be the foci. Then (by Con. Seft. 
 B. I. Prop. II. Cor.) we Ihall have SC or CF z: 
 
 v/RR — CC. Put FP = .V ; then SP =z 2R — x^ 
 and (ib Prop. XXXV ) SP X PF or 2Rx -^ xx ^ 
 BB, and RR — 2Rx -{- xx zz RR -r- BB, and 
 
 R — X - ± \/rR — BB -, whence x =R ± 
 
 \/RR — BB i that is, the greater part FP zi R +
 
 Sea. II. CENTRIPETAL FORCES. 41 
 
 V^RR — BB, and the leffer part SP = R — ■ Fig. 
 
 v/RR — BB. Then in the triangle PCF or PCS, ^'* 
 all the fides are given, to find the angle PCF or 
 PCA. 
 
 Cor. The periodical time in feconds, is 3.1416 
 
 2d 
 
 For arch V^df\ time i" \ : circumference 3.1416 
 
 ^d • T 1 
 
 X id : 3.1416 s/~7 the periodical time in a circle 
 
 whofe radius is d. And by Prop. XVIII. the pe- 
 riodical time is the fame in all circles and ellipfes. 
 
 PROP. XXIV. Froh, 
 ■■ Suppoftng the centripetal force reciprocally as the 2 2, 
 fquare of the diftance ; to determine the orbit which a 
 body will defcrihe ; that is, projected fram a given 
 place P, with a given velocity^ in a given dire^ion PT. 
 
 By Prop, XIII. the body will move in a conic 
 fe6tion, whofe focus is S the center of force. And 
 the line of direction PT will be a tangent at the 
 point P. Let the diftance SP n: J, tranfverfe axis 
 AD = z. / zr fpace a body will defcend at P, in 
 a fecond, by the centripetal force, v zz the veloci- 
 ty the body is projedled with from P, or the fpace 
 
 it defcribes in a fecond. Then \/^df\% the veloci- 
 ty of a body revolving in a circle, at the diftance SP. 
 
 Then (Prop XIV. Cor. \.)v'. s/Tlf: : v/^^IIi: 
 
 \/^z. Whence v^^z zz s/idfz — iddf and 
 
 irvz zz ji^dfz — /\.ddf ; and j\.dfz — vvz zz 4ddf, 
 
 4-ddf 
 whence z zz ■ ,. — zz AD. And PH z= z — 
 
 4aj — vv 
 
 dvv 
 d = »/• ■■■• Therefore if A-df is greater than
 
 42 CENTRIPETAL FORCES. 
 
 Fig. vv, % is affirmative, and the orbit is an ellipfis* 
 22. But if lefler, 2 is negative, and the curve is a hy- 
 perbola, and if equal, *tis a parabola. 
 
 Draw SR perp. to PT, and let SR — p. Alfo 
 draw from the other focus H, HF perp. to PT. 
 Then (Con. Sed. B. I. Prop. X.) the angle SPR 
 r: angle HPF, whence the triangles SPR, HPF 
 arefimilar ; therefore SP (d) : SR (p) : : HP (2: — 
 <^) : HF = --J-p, and (ib. Prop. XXI.) SR X 
 
 HF or ^-^pp = redangle DHA or CB% the 
 
 fquare of half the conjugate axis ; therefore CB =2 
 z — d 
 
 In the triangle SPH, the angle SPH and the 
 fides SP, PH are given, to find the angle PSH> 
 the pofition of the tranfverfe axis. 
 
 Cor. I . ^he periodical time in the ellipfts APDB 
 
 A,ddf 
 
 ZZ3.1416X r 
 
 For 3.1416 v/-7 = periodic time in the circle 
 
 whofe radius is d. And (Prop. XVII.) 2^» : 3. 141^ 
 
 id 
 \/-j : : 2^ : period, time in the ellipfis =1 3.14^^ 
 
 IVd ^^ Tli- ^ Addf 
 
 ^ f 2d ^ 4.df — vvY 
 
 Cor. 2. The laitis return of the axis AD iV =: 
 ppw 
 Idf' 
 
 Cor. 3. Hence the tranfverfe axis and the periodic 
 
 iime will remain the fame^ whatever be the angle of 
 
 dire^cn S?r. 
 
 For
 
 ' B 
 
 M.pO/. 4'2.
 
 r^
 
 Seft.n. CENTRIPETAL FORCES. 4^ 
 
 For no quantities but ^, /, and v are concerned s Fig. 
 all which are given. 22. 
 
 Scholium. 
 Some people have dreamed that there may be a 
 fyftem of a fun and planets revolving about it, 
 within any fmall particle of matter ; or a world in 
 miniature. But this cannot be ; for though mat- 
 ter is infinitely divifible ; yet the law of attraftion 
 of the fmall particles of matter, not being as the 
 fquares of the diflances reciprocally, but nearer the 
 cubes ', therefore the revolution of one particle of 
 matter about another, cannot be performed in an 
 cUipfis, but in fome other curve ; where it will con- 
 tinually approach to or recede from the center ; 
 and fo at laft will lofe its motion. Such motion as 
 thefe can be nothing like that of a fun and planets. 
 
 PROP. XXV. 
 
 If a body revolves in the circumference of a circle 24. 
 ZPA, in a rejijling medium^ whofe denjity is given. 
 To find the force at any place P, tending to the cen- 
 ter C i as alfo the time^ velocity^ and reftfiance. Sup* 
 ■pofing the refinance as the fquare of the velocity. 
 
 Draw PC, and dp parallel and infinitely near it, 
 cutting the tangent Vd in d. And put CZ zr r, 
 ZP =::: 2;, time of defcribing ZP z= /, velocity at 
 P zr 1;, refinance — R, / zr force at P, ^ zz force 
 of gravity at Z, r zz velocity in Z. And let a 
 body moving uniformly with the velocity i, thro* 
 the fpace i, in the time i, meet the refiftance i 
 in the medium. And let a body defcend thro' the 
 fpace ^, by the force g at Z, in the fame time i. 
 
 I. By the laws of uniform motion, the Ipace is 
 as the time X velocity. Whence i (fpace) : i X i 
 
 (time X vel.) i i z '. vt zi z^ whence / =z — • 
 
 2. By
 
 ^ CENTRIPETAL FORCES. 
 
 - . 2. By the nature of the circle, dpzz— :=z • 
 
 3. By accelerated motion, the fpace is as the 
 force X fquare of the time ; whence^ X 1^ (force 
 
 X time*) : a (fpace) : : /// : dp or : : zrf : vo 
 
 2arf . J 
 
 • And cc — zar. 
 
 g 
 
 4. The velocity generated (or deftroyed) is as the 
 force X time ; therefore, gX 1 (force x time) : 2a 
 
 2aR} ^dRz , 
 (velocity) : : R/:-^i; = = -^,and — 
 
 2dRz 
 vv = -J-' 
 
 5. The refiftance is as the fquare of the veloci- 
 ty, whence i* (vel/) : i (refiftance) ::vv:Rzz vv, 
 
 2aRz 2avvz 
 
 Therefore — vv zz — And — 
 
 i g 
 
 ' 2az 2az 
 
 — — » whence - — log : v zz ' and correCt- 
 
 V i ^ g 
 
 c 2az 
 cd, log : — zz 
 
 Again, fince —_- zz vv, —^ zz vv zz — 
 
 zavvz , • 2V-JZ 4arfz f 
 
 ■ 5 and f =: — zz » and — ^ 
 
 zz. — » and — log :/ — — , and correded, log : 
 
 g o 
 
 £ _ 4f5 
 /"■ ^ . 
 
 Alfo i zz— zz ml^, and t zz — ■> and cor- 
 
 V 2avv 2^1; 
 
 refted / iz -^ — ^ . ^ 
 
 2av zac Cor*
 
 Sea. IL CENTRIPETAL FORCES. 45 
 
 Cor. I. Hence v zz number belonging to the Icga- Fig. 
 
 rithm : log : c • And f-zi number belonging to 
 
 the logarithm : log :g --^ ■ — ' 
 
 Cor. 2. Therefore the logarithms of v andf, each 
 of them fever ally decreafes equally^ in defcribing equal 
 fpaceSy ad infinitum. And therefore at every revolu- 
 tion, the log : of v is equally diminijhed, and Ukewife 
 that of f But the body will revolve for ever, for 
 when V is o, / will be infinite. 
 
 Cor. 3. Hence if the central body at C, was fa 
 diminifhed that its log : may decreafe equally in defcrib- 
 ing equal fpaces, or in each revolution, after the man- 
 ner as before-mentioned \ then the body will perpetu- 
 ally revolve in a circle, in a medium of uniform den- 
 fity. 
 
 SECT.
 
 t46] 
 Fig. 
 
 SECT. III. 
 
 ^he ^notion of three bodies aSling upon 
 one another \ the perturbuting forces 
 of a third body, The motion of bo- 
 dies round an axis at refiy or having 
 a progrefftve motion^ and other things 
 of the fame nature, 
 
 PROP. XXVI. 
 
 25. If a body he proje5led from A, in a given direHion 
 AD, and be attracted to two fixed centers S, T, not 
 in the fame plane with AD ; ^ the revolving triangle 
 SAT, drawn thro* the moving body^ Jhall defcrihe e- 
 qual folids in equal times, about the line ST. 
 
 Divide the time into infinitely fmall equal parts ; 
 it is plain that equal right lines AB, BC, CD, &c. 
 would be defcribed in thefe equal times ; and con- 
 fequently that all the folid pyramids STAB, STBC, 
 STCD, &c. are equal, which would be defcribed 
 in the fame equal times ; if the moving body was 
 not a6led on by the forces S and T. 
 
 But let the forces at S and T, aft at the end of 
 the feveral intervals of time ; as fuppofe the force 
 T to a6t at B in direction BT ; fo that the body, 
 inftead of being at C, is drawn from the line BC, 
 in the direftion CF, parallel to BT. And in like 
 manner it is drawn from the line BC, by the force 
 S, in diredlion CE parallel to SB. And therefore, 
 by the joint forces, the body at the end of the time, 
 mult be fomewhere in the plane ECF parallel to 
 
 SBT,
 
 Sea, III. CENTRIPETAL FORCES. 47. 
 
 SBT, as at I. But (Geom. VI. 17.) the folid Fig. 
 pyramids S TBI and STBC, are equal ; being con- 25. 
 tained between the parallel planes ECF and SBT, 
 and therefore have equal hights ; whence STBI =: 
 pyramid STAB. 
 
 In like manner continue BI, making IK = BI ; 
 and in the next part of time, the body would ar- 
 rive at K, defcribing the pyramid STIR equal to 
 STBI. But being drawn from the line IK, by the 
 forces S, T, in the diredions KL, KN, parallel 
 to IS, IT ; the body will be found at the end of 
 the time, fomewhere in the plane LKN parallel to 
 SIT, as fuppofe at O, and then it will have de- 
 fcribed the folid STIO - STIK :- STBI =: py- 
 ramid STAB. 
 
 And in the fame manner producing lO to P, till 
 OP zr 01. Then the body, attraded from O, by 
 the forces S, T, will defcribe another equal pyra- 
 mid. And fo it v/ill continue to defcribe equal py- 
 ramids in equal times; and confequently the whole 
 folids defcribed are proportional to the times of de- 
 fcription. 
 
 Cor. I. PFhen the number of Uneoltg AB, BI, lO, 
 t^c. is iricreafed, and their magnitude diminijhed^ ad 
 infimtum ; the orbit ABIO, becomes a curve. 
 
 Cor. 2. Any line AB is a tangent at A, BI at B, 
 i^c. A, B, (^c\ being any 'points in the orbit. 
 
 Cor. 3. But the orbit ABIO is not contained in 
 me plane^ except in fome particular cafes. 
 
 For that the orbit may not deviate from a plane ; 
 the forces on both fides thereof, cugiit to be alike. 
 
 PROP.
 
 4% CENTRIPETAL FORCES. 
 
 Fig. 
 
 PROP. XXVII. 
 
 26. If the body T revolves in the orbit TH, about the 
 body S at a great dijlance, whiljl a lejfer body P re- 
 volves about T very near ; and if C be the centripe- 
 tal force of S aSting upon T. Then the dijiurbing 
 
 3PK 
 force of S upon P is — "5j~C. Suppofing PK pa- 
 
 PT 
 
 rallel^ and KT perp. to ST. y^;/^ -^C zz the in- 
 
 creafe of centripetal force from P towards T. 
 
 Let ST —r,VT—a,VY^—y,g — force of 
 gravity, h zi fpace defcended thereby in time r. 
 J — the fpace defcended in the time i, by the force 
 C. ^ rr periodic time of T about S, and t :=: per. 
 time of P about T. y — centripetal force of T at 
 P, TT zz 3.1416. 
 
 Since attraction is reciprocally as the fquare of 
 the diftance, then force of S afting at T : force of 
 
 S afting at P : : -^^ : ^p 
 
 rr ^__y 
 
 r + 2jy, nearly. And force of S adling at T : to 
 difference of the forces : : r : 2y; that is, r : 2y : : 
 
 C : — C zi difference of the forces : and this is 
 
 T 
 
 the fingle force by which P is drawn from the or* 
 bit QAZ in diredion KP or PS. 
 
 But fince the motion will be the fame, whether 
 the fmgle force PS aft in the direcftion PS -, or the 
 two forces PT, TS aft in the direftions PT, TS ; 
 fubftitute thefe two for that fingle one •, therefore 
 proceeding as before, the force of S adling at T : 
 
 force of S afting at P : : — ; • And force 
 
 ^ rr r—y 
 
 of
 
 Sea. m. CENtRIPEtAL FORCES. 4^ 
 
 of S ading at P in diredion PS : force of S a6l- Fig. 
 ing on P in diredion of TS : : PS : TS : : r — y\ 26. 
 
 r : I -: fZIZ' Therefore ^;f ^^«(7, force of S afting 
 
 at T : force of S ading on P in diredion TS : : 
 i I j_ I i I 
 
 t^ ' ^mir^' * * ri ' r^ — ^ry ' '~r ' r — ^y' ' ^ ' 
 
 r + 3JJ, nearly. And the force at T : difference 
 
 of the forces ::r: ^', or r: ^y i iC : ^ C ^ dif- 
 
 turbing force of P, afting ih direflion parallel to 
 TS. And PK (y) : PT (a) : : increafe of the dif- 
 
 /y \ d 
 turbing force in direftion PK ( — C ) : — C, the ad- 
 dition of the centripetal force in direftion PT. 
 
 2V 
 
 For when the difturbing force was —C, there was 
 
 ho addition of centripetal force at T, but a dimi- 
 nution thereof; as appears by the following Corol. 
 
 Cor. I . The fimpk difturhing force, whereby P is 
 
 drawn towards S, is ~ -^^C. And the diminution of 
 
 1) 
 centiipetal force ofV towards T, is z= - C. And the 
 
 accelerating force at P in the arch PA, is zz — C. 
 
 Putting z zzfine of 2PQ, v zz ver fed fine of iVQ^ 
 
 For kt X zz PK, and draw Kl perp. to PT ; 
 
 then b, fimllar triangVs, PT (a) : PK (y) : : 
 
 PK : PI : : force PK (jC^ : force in diredionlP 
 
 2yv_ v^ 
 
 or TP z: -^C zi - C. 
 cir r 
 
 Aifo PT (a) : IK (x) : : PK : KI : : force PK 
 
 E (.zy
 
 50 CENTRIPETAL FORCES. 
 
 ^f * (-C) : force in diredion KI or PA z= ^ C =: 
 
 -C. By Trigon. B. I. Prop. II. Schol. 
 
 q 
 Cor. 2. 'The dijiu7'bing force at P is zz rrTT^ i 
 
 59t 
 being the fine of the difiance from the quadraturey P 
 
 the rnoon, S the fun. 
 
 ttr 2V 7tty 
 
 For (Prop. V.) C = —-7, and ^-^Czz^-fy- 
 
 ^^^ (becaufe - =: - ) zz —7 r nearly. 
 
 Cor. 3. If S he the fun, P a body in the equinoc- 
 tial of the earth ; the difturhing force at P is zz. 
 
 12852000 
 For when P is at the moon's orbit, the force js 
 
 ^y ; but ^ = 60 X 607, or y = -^—^, therefore 
 
 qg 
 
 the force becomes — tz. — 7 — ' and at the earth is 
 59I-X3600 
 
 59'-X 6o5 
 
 Cor. /i^. If S he the moon, P t7 /^oi)' o« //??^ cqui- 
 nofiial of the earth. The difturhing fot ce at P is zz 
 
 2880000 
 
 For the general perturbating force was -^ C, and 
 
 here C muft be the centripetal force at the moon. 
 Now the centripetal force of the earth, at the dif- 
 
 tance of the moon is j—.g' And the moon being 
 
 40 tim€S kfs than the earth, the centripetal force 
 
 of
 
 Sea. III. CENTRIPETAL FORCES. 
 
 of the moon, at the fame diftance, is ^-r p--. ^^' 
 
 ' 40x60* * ' 20. 
 
 put this for C, then the force of the moon upon 
 
 the equinodial, is — X ■ — -tt"! = 7 — 7-r 
 
 ^ ' r 40 X 60* 60a X 40 X 60* 
 
 20 X 40 X 60* 
 
 Cor. 5. The dijittrhing force of the fun ^ to that of 
 the moon^ upon the equinohial^ is as i to 4.. 4.6. 
 
 For thefe forces are as — n and 
 
 12852000 . 2S80000 
 or as 288 to 1285, or as i to 4.46. 
 
 Cor. 6. If f he the a-pparent diameter^ tind d the 
 denfity of the perturhating body. Then the dijiurbing 
 force will always he as df'y. 
 
 3y y^ ' 
 
 For that force is — C pr as — * Let its diame- 
 r T 
 
 ter, — : ^, M :z: its quantity of matter. Then C 
 
 M dh 
 IS as — i that is, as Therefore the difturbing 
 
 YY YT 
 
 db'y 
 force IS as — p, or as f/y X/'. 
 
 Cor. 7. //" P he a point in the equator of the 
 earth, S the fun. 
 The centrifugal force of P : 
 is to the fertm'baiing force PT : : 
 As the fquare of the earth's periodical time about the 
 
 fun pp : 
 to the fquare of the eartlfs ■periodical time about iu 
 axis tt. 
 
 Let / — time of revolution of the earth round 
 its axis J then t : z-rta ^circumference} : : 1' : 
 
 -y- zi arch defcribed in one fecond j and the verf- 
 
 E 2 ed
 
 52 CENTRIPETAL FORCES. 
 
 Fig. /^TTTraa l-mra 
 
 25. ed fine iz ^^ ^ ^^ = — ^ =: afcent or defcent by 
 the earth's centrifugal force. But forces are as 
 
 2 TTira 2 TTtrttE 
 
 their effeds, whence b : g : : —-- (afcent) : — jr^ 
 
 the centrifugal force itfelf. But the perturbating 
 
 f.C - ^ 
 r ~ rb 
 
 a asg 
 
 force is — C =. —r' Whence the centrif. force 
 
 l-mrag asg iTTTT s 
 
 perturbatmg force ■ ■ -JJT ' 7h ■ • IT 'T '■ 
 
 2-mrr Z-mrr 
 
 2tnrr : tts : : — - — : //. But i= pp. Vor^irs 
 
 i-nr 4.vnrr 2wr 
 
 I : : 27rr : p = . — > and pp zz 
 
 2rs s 
 
 Cor. 8. Hence the body V is accelerated from the 
 quadratures Q, Z, /^ theftziges A, B ; ^»^ retard- 
 ed from the fiziges to the quadratures. And moves 
 f after, and defcribes a greater area^ in the fiziges 
 than in the quadratures, 
 
 PROP. XXVIII. 
 
 I^he fame things fuppofed as in the laft Prop, the 
 linear error generated in P in any time, is as the dif- 
 turbing force and fquare of the time. And the an- 
 gular error, feen from T, will be as the force and 
 fquare of the time directly, and the di fiance TP reci- 
 procally. 
 
 For the motion generated in a given part of time, 
 by any force, will be as that force j and in any- 
 other time as the force and the fquare of the time. 
 The motion fo generated is the linear error of P, a* 
 it is carried out of its proper orbit, by the force 
 
 - C. And that error, as feen from T, is as the 
 
 T 
 
 angle
 
 Sea. III. CENTRIPETAL FORCES. 53 
 
 angle it is feen under ; and therefore is as that linear Fig. 
 error, divided by the di (lance TP ; and therefore 26. 
 is as the force and fquare of the time, divided by 
 the diftance. 
 
 Cor. I . the linear error generated in one revolu- 
 tion of P, is as the dijiurhing force and fquare of 
 
 oca 
 the periodical tim-Sy — //. y^nd the angular error in 
 
 one revolution is as the force and fquare of the perio- 
 die time divided by the dijla?ue. 
 
 Cor. 2. The mean linear error of P in any given 
 
 a 
 time, will he as the force and periodical time, — Qt, 
 
 And the mean angular one, as the force and periodical 
 
 time, divided by the dijlance. 
 
 For let the given time be i ; then / (time) : 
 
 aC . . . aCt , 
 
 — tt (whole error) : : i : — > the error m the 
 
 given time. 
 
 Cor. 3. The mean lineal error in any given time, i$ 
 as TP and the periodical time of P dire5lly, and the 
 fquare of the periodical time of T reciprocally. And 
 the mean angular error, as the periodical time of P 
 dire^ly, and the fquare of the periodical time of T 
 reciprocally, 
 
 r at . at 
 
 For (Prop. V.) C is as —> and — G is as — X 
 
 r at t 
 
 ■—- or —-• And the angular error as — ■ 
 pp pp ^ pp 
 
 Cor. 4. In any given time, the lineal error is as 
 TP and the periodical time of P dire£fly, and the 
 cube of ST reciprocally. And the angular error as 
 the periodical time of P dire^ly, and the cube of ST 
 xeciprocally, 
 
 E 3 Fop
 
 -^^ CENTRIPETAL FORCES. 
 
 Fig, , r ^^ ' ^^ 
 
 j| For (Prop. XVII.)/!p is as r\ therefore —is as-y 
 
 Cor. 5. 5"^^ /f>f^r ^rr^r in a given time is as 
 1 -? 
 
 flC, «K^ the angular error as fIC, 
 r r 
 
 at ^ 
 For / is as ^\ and ■— C ^j — C. 
 
 Cor. 6. And univerfally, the angular errors in the 
 whole revolution of any fatellites ; are as the fquares 
 of the -periodic times of the fatellites dire£lly^ and the 
 fquares of the ■periodic times of their primary planets 
 reciprocally. And the mean angular errors are as the 
 periodical times of the fatellites^ divided by the fquares 
 of the periodic times of their primary planets. 
 . For by Cor. i. the angular error is as the force 
 ^nd fquare of the time divided by the diftance \ 
 
 1 . C« // . r ^ • '^ \ 
 
 that IS, as — x ~ ; that is, (becauie C is as — ] 
 
 r a ^ ppf 
 
 tt 
 as — • The reft is proved in Cor. 3. 
 
 PROP. XXIX. 
 
 27. If a fpheroid AB revolves about an axis ST in 
 free fpace^ which axis is in an oblique fituation to the 
 fpheroid -, the fpheroid unll^ by the centrifugal force, 
 i?e moved by degres into a right pofition ab ; and af- 
 terwards by its libration., into the oblique pofition a (3. 
 And then will return back into the pofitions ah, AB ; 
 dpd fo vibrate for ever. 
 
 Let C be the center of the fpheroid •, D the cen- 
 ter of gravity of the end ICLB •, E that of the 
 end ICLA ; D^, 'Ee perp. to ST. Then the cen- 
 trifugal force of the end CB, fuppofing it to z8i 
 
 wholly
 
 Sea. III. CENTRIPETAL FORCES. 55 
 
 wholly at D, in diredion dD, having nothing to Fig. 
 oppofe it, will move the end CB from B towards 27. 
 ^, with a force which is as ( d. And at the fame 
 time, the centrifugal force of the end CA, ading 
 in diredion fE, will move the end CA from A to- 
 wards a, confpiring with the motion of the end 
 CB ; by which means it will by degrees come into 
 the pofition al?. And then by the motion acquir- 
 ed, it will come into the pofition a ^, making 
 the angle SC « — SCB. And the motion being 
 then deftroyed, it will return back, by the like 
 centrifugal forces, adting the contrary way -, and 
 be brought again into the pofitions al>, and AB j 
 and continue to vibrate thus perpetually. 
 
 Cor. Hencef if the axis of the earth is not precife* 
 ly the fame as the axis of its diurnal rotation -, the 
 earth will have fuch a libration as is here defcribed^ 
 hut exceeding finall. I'his is fupfofin^ it a folid bo- 
 dy ; but if it was a fluids it would by the centrifu* 
 gal force^ form itfelf into an oblate fpheroid, 
 
 PROP. XXX. Prob. 
 
 If a globe AVQQ^in free fpace^ revolve about 2S, 
 the axis SCT, in direction ADB ; and if any force 
 applied at V, the end of the radius CV, ai^ts by a 
 /ingle impulfe in dire^ion VG perp. to CV, in the 
 plane VCD. To find the axis about which the globs 
 (hall afterwards revolve. 
 
 Suppofe the great circle VBQ/K perp. to the 
 line of diredion VG •, and if VH, VI be 90 de- 
 grees ; it is plain, if the firil motion was to ceafe ; 
 the globe by the impulfe at V, would revolve round 
 the axis IH, which by the firft motion was round 
 the axis PQ^ Therefore by both motions toge- 
 ther it will move round neither of them., Now 
 
 E 4 fmce
 
 §6 CENTRIPETAL FORCES. 
 
 Fig. fince a point of the furface moving with the greats 
 28. eft velocity about ST, will move along the great 
 circle ADB •, and a point having the greateft velo-. 
 city about IH, moves along the great circle VDE. 
 Therefore a point that will have the greateft velo- 
 city, by the compound motion, will alfo be in a 
 great circle pafling thro' D. Therefore in the 
 great circles DB, DE, take two very fmall lines 
 Dr, Do, in the fame ratio as the velocities in AD, 
 and VD ; and complcat the parallelogram Dopr, 
 Then thro' D and p draw a great circle KD^L ; 
 and a point having the greateft motion, arifing from 
 a compofition of the other two motions, will move 
 along KD^L. Therefore finding F, R the poles 
 of the circle KDj)L, FR will be the new axis of 
 revolution, or the axis fought. And the velocity 
 about the axis FR will be proportional to Dp ; 
 VBQA being always fuppofed perp. to GV, or to 
 the plane DVC. 
 
 - Note, if you fuppofe an equal force applied at 
 E, in diredion contrary to GV, it will by that 
 means keep the center C of the globe unmoved, 
 and will likewife generate twice the motion in the 
 globe. 
 
 Cor. I . The greater the force is that is applied t& 
 V, the greater the dijlance PF is, to which the pole 
 is removed. And if fever al impulfes he made fuccef- 
 Jively at V, when V is in the circle APB, the pole F 
 will be moved further and further towards H, in the 
 circle APB. 
 
 For feveral fmall forces or impulfes have the 
 fame effeft as a fmgle one equal to them all. 
 
 Cor. 2, If the force a5l at P, in direction perp. t» 
 the plane CPB •, and Dr, Do be as the velocities along 
 DB and DQ. The great circle KDE (paffing thro* 
 D end p) is the path of the point D ; and its pole ¥, 
 sr axis of revolution RF ; the pole being tranjlated 
 
 from
 
 Sea. m. CENTRIPETAL FORCES. 57 
 
 from VtoV. And if the impulfe he exceeding fmall. Fig, 
 PF will alfo be exceeding fmall. 29. 
 
 Cor. 3. If the force at P always a5ls in parallel 
 dire^ions, whilfi the globe turns round. The pole 
 will make a revolution in a fmall circle upon the fur- 
 face of the globe, in the time of the globe's rotation^ 
 and the contrary way to the globe's motion. 
 
 For let a fingle impulfe at P tranflatc the pole to 
 F -, and afterwards when the globe has made half a 
 revolution, and the point P is come to p ; then if a 
 new impulfe be made at F, the pole will be trans- 
 lated top which is now P ; that is, it will ^e mov- 
 ed back to its firft place on the globe. So that 
 in any two oppofite points of rotation, the place of 
 the pole is moved contrary ways, and fo is carried 
 back again the fame diftance. And fince the globe 
 revolves uniformly, if the force a6t uniformly, it 
 •will move the pole all manner of ways, or in all 
 manner of direftions upon its furface ; that is, it 
 will defcribe a circle, which will end where it be- 
 gun. And in defcribing this polar circle, the mo- 
 tion will be contrary to the motion of the globe; 
 for fuppofe PFB an immovable plane. If the globe 
 ftood ilill, the pole would move in a great circle, 
 in the plane PFB. But fince all the points of the 
 globe which come fuccefUvely to the plane PB, are 
 not yet arrived at it, but are fo much further fhort 
 of it, as PF is greater ; 'tis evident all thefe points 
 will lie on this fide the plane PB. And as any 
 fixed point will defcribe a circle on the moving 
 globe, contrary to the motion of the globe ; fo will 
 a point that is not fixed, but moving in the plane 
 PFB, likewife defcribe a circle (or fome curve) 
 contrary to the motion of the globe. Or fhorter 
 thus, fuppofe the globe to Hand ftill, and the di- 
 redlion of the force to move backward, then the 
 relative motion wiU be the fame as before; and 
 
 then
 
 5S CENTRIPETAL FORCES, 
 
 f igi then the pole F will move backward too, as it will 
 2€f. follow the force, being at right angles to it. 
 
 Cor. 4. Since the pole hy one impulfe is tranjlated 
 to F *, the new pole F is therefore another point ef 
 the material globe, diftin^l from P. Jnd the particle 
 P that was before at reji, will now revolve about the 
 panicle F at rejl. 
 
 For the new pole F is that particle of the globe 
 which happened to be revolving in F, when the 
 impulfe was made at P. The matter of the great 
 circle ADB dees not come into the circle KDL, 
 but only the point D of ir. For when the force is 
 impreHed, the other particles M, N, by the com- 
 pound motion, will be made to revolve in the di- 
 reftions M;;^, N«, parallel to D/> j and therefore 
 will defcribe leffer circles about F; whilfl: only D 
 defcribes the great circle KDL. 
 
 Cor. 5. What is demonflrated of a fphere is true 
 clfo of an oblate fpheroid, whofe axis is PQ^; and 
 the force imprcffed at P, a^ing in direSlion perp. to 
 CPBj or parallel to CD the radius. r- 
 
 Cor. 6. But if the force at P aEi in direSlion con- 
 trary to the foregoing {as in cafe of an obiong fphe- 
 roid) ; the pole of rotation will be moved from P to- 
 wards A, contrary to the way of the other motion. 
 
 Cor. 7. Jnd in general the pole P will always be 
 iftoved in a direction perp. to that of the power j and 
 towards the ja'me way as the fpheroid revolves. 
 
 Cor. S Hence after every half rotation of the 
 globe round its axis, the places upon the globe change 
 their latitude a little \ which, after an entire rotation 
 return to the fame quantity. But this variation is f9 
 trifling, as to come under no ohjervation. 
 
 This is evident, becaufe the pole is altered \ and
 
 '\b 
 
 ^-"^^ 
 
 JE./w. 3<«-'.
 
 Sea. III. CENTRIPETAL FORCES. 59 
 
 of confequence, the diftance therefrom is altered Fig. 
 in all places, except in the great circle FCR. 29. 
 
 PROP. XXXI. Proh. 
 
 Let AB he an oblate fpheroid, whofe axis is PC j Z^^ 
 find let it revolve round that axis, in the order ADB, 
 which is its equino£lial\ and if any force a5i at P, 
 in direction PG perp. to PC, and in the plane P O C, 
 which moves flowly about y in the order ADB. *To 
 find the motion generated in the fpheroid. 
 
 Let EL be an immovable plane like the eclip- 
 tic, in which the center C of the fpheroid always 
 remains. ON another plane parallel to it. Erecb 
 CM perp. to thefe planes -, and make the angle 
 MCN n the angle B^^C, in which the equinoftial 
 B^ cuts the ecliptic CL. Suppofe the fpherical 
 furface OVN to be drawn, whole pole is M •, and 
 produce CP to cut it at R, in the circle ORN. 
 Then PCD and RCM are in one plane, and both 
 of them perp. to P;a and PtC. Now to find the 
 motion of the axis of the fpheroid. Here OVN is 
 the upper fide of the furface. 
 
 This Prop, differs from the laft in feveral re- 
 fpefbs. The laft Prop, regards only the motion of 
 the pole upon the furface of the globe, and that 
 js caufed by a motion which is generated in the 
 globe itfelf. But in this Prop, we confiderthe mo- 
 tion of the axis CR in the fixed fpherical furface 
 OVN i which always proceeds in one diredion, as 
 long as the moving force keeps its pofition. In 
 the laft Prop, the motion of the globe round its 
 axis is performed in a very fhort time \ but here 
 the revolution of the force, in the moving plane 
 P G C, is a long time in its period. 
 
 Now by the laft Prop. Cor. 7. the force PG will 
 always move the axis of rotation CPR in a direct 
 
 tioa
 
 ^0 CENTRIPETAL FORCES. 
 
 Fio-. tion perp. to PG. Therefore fiippofe the plane 
 30. P G C to revolve (lowly round PC, and we fhall 
 find that in the beginning of the motion, when O 
 or F is at i^, then the plane (PgC or) P£:C will 
 be perp. to the plane CRM, and at that time the 
 motion of R will be directed from R towards M. 
 And when that plane comes to the pofition PF o C, 
 the motion of R will be direded to fome place be- 
 tween N and V. And when it is got to the tropic 
 D, then R's motion is diretted along the circum- 
 ference RV ; for then P o C coincides with CRM. 
 But when P G C arrives at /, the motion will be 
 direfted from R to fome point / without the circle. 
 And laftly, when P G C is at the other interfeftion 
 T, beyond B ; the motion of R will be direfbed to 
 m oppofite to M. The refult of all which is, that 
 the pole R will defcribe fuch a curve asRi234-, 
 and then the fame force begins again at — •, which 
 being repeated, another fimilar figure 456. is de- 
 fcribed by R •, and fo on for more. The fame 
 force I fay is repeated, for when the plane PgC 
 comes on the other fide of the globe, the force 
 a6ls the contrary way, and therefore 'tis all one as 
 if it aded on the firft fide of the globe. 
 
 Itmuft be obferved, that as R moves thro' 1234,, 
 the interfeftion £: gradually moves towards E. And 
 as to the force PG, it may be fuppofed variable, 
 at different pofitions of the plane PgC. And 
 according to the quantity of force in the feveral 
 places, different curves (1234) will be defcribed. 
 
 Cor. I. Hence it is evide?it, that the inclination of 
 the planes ADB and ECL, is greatejt when PgC 
 fajjes thro* £: and T . And leafi when it pajfes thro^ 
 the tropic D. And that the inclination deer eafes from 
 the node ^ to the tropic D, and increafes from the 
 tropic to the node. 
 
 For
 
 Sea. III. CENTRIPETAL FORCES; 6i 
 
 For when P0C is in -, R is fartheft from M ; Fig. 
 but when it is in D, R is at 2, and its direction ^o, 
 is parallel to R4, and then 2M is leaft. 
 
 Cor. 2. ^fier a revolution of the 'plane PFoC 
 {in which the force always a£ls)^ the inclination comes 
 to the very fame it was at firjl. 
 
 For at any two points F, /, equidillant on each 
 fide from the tropic ; the force is diredled contrary 
 ways, from and to the circle RV ; and therefore 
 the motion, in the curve R1234, being alfo equal 
 and contrary, from and towards RV, they mutu- 
 ally deftroy one another •, and therefore after a re- 
 volution, or rather half a revolution, the pole R 
 is brought back to the circle RV, and then the 
 angle RCM is the fame it was at firft. 
 
 Cor. 3. The motion of the pole R, reckoned in the 
 (ircle O VN, is always from R towards V, then thro* 
 N, O, and R. 
 
 For tho* the motion of R towards and from M, 
 in the line Mw, in one revolution, is equal both 
 ways; and fo R is always brought to the circle 
 again ; yet the motion confidered along the circle 
 is always in the order RVN. Thus it goes thro* 
 the curve 123 to 4, fo that after half a revolution 
 of PqC, it is advanced forward in the circle RV, 
 the length R4. 
 
 Cor. 4. The motion along the circle is fometimes 
 f after and fometimes flower. At 2 it moves faftefi 
 of all ', at R and 4, it moves floweft, or rather is 
 fiationary for a moment. 
 
 ' Cor. 5. The pole R, and the nodes move the con- 
 trary way about y to what AB revolves. 
 
 Cor. 6. If the force PG ftands ftilk the pole R 
 ijoill ftill move backwards as before i and that in a 
 
 -right
 
 6z CENTRIPETAL FORCES. 
 
 Fig. right line, or rather, a great circle. And if PG moves' 
 
 30. backward thrd BDFA j the 'pole R will Jiill go backr 
 
 ward i but then the curve R24, will be concave tOr 
 
 wards M, like 87R, being contrary to the other 
 
 where the force moved forward. 
 
 Cor. 7. But in an oblong fpheroid, where the force 
 a5ls in dire£li:n GP, quite contrary to the other ; 
 R will defer ibe the curve R78, without the circle 
 ORV ; every far tide of it in a contrayy dire^ion to 
 thefe of R24. And therefore the pole ,R, and the 
 nodes T ^'nd £5 will move the fame way about as ADB 
 revolves, end contrary to what they dq in an oblqfp 
 fpheroid. 
 
 For the force being diredbed die contrary way j 
 of confequence the motion muft be fo too. 
 
 Cor. 8. And in an oblong fpheroid, if the force GP 
 move the contrary way about ; yet the pole R will 
 fiill move forward. And the curve defcribed by R, 
 'will have its convexity the contrary way. 
 
 Cor. 9. Hence if the quantities and proportions of 
 thefe forces, in different places be known ; it will not 
 be difficult to delineate the curve R1234, upon the 
 fphericcl fur face OVNM. 
 
 PROP. XXXn. ?rob. 
 
 51- If a planet {or the moon) move in the orbit ATE/, 
 round an immovable center C, whofe plane is inclined 
 to the plane of the ecliptic AQE ; and a force aEls 
 upon it in liries perp. to GZ, and parallel to the eclip^ 
 tic, directed always from the plane G7L to either Jide. 
 To find the motion of the nodes A, E j una the varia^ 
 lion of the orbit's inclination PAO. 
 
 Let ATZE be half the orbit raifed above the 
 ecliptic AQ, AE the line of the nodes j T, /, 
 
 the
 
 Sea. III. CENTRIPETAL FORCES. ^j 
 
 the tropics. Draw CM perp, to the ecliptic AQE, Fig. 
 and CR perp. to the orbit AT E. Round M as 3U 
 a pole defcribe a fpherical lurface RVNX-, then 
 RC will be the axis of the orbit, and MC of the 
 ecliptic, or circle AQE. Thro' the points T, C, 
 M, draw the plane RMC •, and thro' A, C, M, 
 another plane cutting the circle RNF in X andHj 
 then RF is perp. XH, and the cncle RNX paral- 
 lel to GEQ. Note, RNX is the upper fide of the 
 furface. 
 
 Let the planet be at P, and let P2 be the fpace 
 it defcribes in any fmali time, and the line Pi the 
 fpace it would be drawn thro' in the farre time, by 
 the force a<5ting from the plane MGZ. Compieat the 
 parallelogram Pi<!3, and P3 will be its direcflion 
 by the compound fotce. Now as the line Pi is 
 parallel to the ecliptic, 'tis plain the point 3 will be 
 below the plane ot the orhit -, and the plane CF2 
 will be moved mto the pciition CF3, revolving 
 about Ct' ', conftquenciv the axis RC will be mov-i 
 ed in a direftion perp. to CP. And the pole R 
 will be moved to lome point between F and H. 
 This being duly attended to, t^ie motion of the 
 pole R will be known for all the places of P in 
 the orbit GATZ For about G the motion of R 
 is direded pctrp. to Mn ; at A it moves perp. to 
 MX, or in dire6tion RM At T it moves parallel 
 to Mi-1, or in the cui-ve RV. Approaching to Z, 
 it moves perp. from 1V1«. So that in the pafTagc 
 of the planet P^ from G to Z, thro' GAPTZ ; 
 the pole R of its orbit, moves thro' the curve. 
 Ri 1 .4. But m the other half of the^ orbit ZE/G, 
 as the fo-ce i§ direded the contrary y/ay from tjbe 
 plane CtZMN, the pole R will return back at 4, 
 and defcnbe a fimilar curve ^c^G'j'i. So that when 
 tije planet P has m^de one revolution, the pole of 
 its orbit R v/ill be found at 8. But in this Dofition 
 ^ cf rhe noues, the point 8 will be within tlie fphe- 
 rical
 
 64 CENTRIPETAL FORCES. 
 
 Fig. rical furface RHNX, not reaching the periphcrj^ 
 31. RV. For the feveral parts of the curve be- 
 ing defcribed in all directions in refpefb of the 
 line nN j the points R, 4, will be equidiftant from 
 »N, and likewife 4, 8, for the fame reafon. 
 
 Now fuppofe the force and the plane GZN/7 to 
 to revolve about the axis CM in the order GATE. 
 Then after it is come to fuch a pofition, that the 
 afcending node A is as far on the other fide of G, 
 fuppofe at a -, then the pole R will be as far on the 
 other fide of V, fuppofe at r ; and being alfo as far 
 from N», on the fame fide; the curves (12468) 
 will approach VH there, by the fame degrees as 
 they receded from it at RV. And therefore the: 
 pole R will by degrees be brought to the circle 
 again. Thus in every two correfpondent points on 
 each fide V, the forces and their effedls balance 
 one another, and R will be at the fame diftancc 
 from the circle RVH. And therefore after half a 
 revolution of the plane GZ to the nodes, the angle 
 RCM, and confequently the inclination of the or- 
 bit, comes to the fame as at firft. And likewife as 
 the pole R moves foi-ward or backward in the 
 circle RVH •, the motion of the nodes A, E, will 
 be forward or backward. 
 
 Cor. I . In ibis pofition of the nodes at A and E, 
 the inclination of the orbit ATE will he diminifjed: 
 every revolution of P. But on the cppofite fide at a^ 
 the inclination increafes every revolution of Ir*. 
 
 For the points 4, 8, come nearer and nearer to 
 M, and the contrary at r. 
 
 Cor. 2. JVhen the nodes are at ^^ E; the inclina- 
 tion decreafes \ when the planet is in GT or Xt •, and 
 increafes in TZ and tG. 
 
 For R moves to 3, whilft P moves thro* GT. 
 At 3 it is at its neareit diftancc to ivl ; from 3 to 
 
 4R
 
 Sea. III. CENtRIPETAL FORCES. 6^ 
 
 4 R recedes from M, whilft P moves thro' TZ. Fig. 
 And the like on the other half of the orbit. 3 1 , 
 
 Cor. 3. When the planet is in GA and ZE, the 
 nodes go forward. But in AZ and EG, they go 
 backward. 
 
 For whilft R pafles thro' Ri, its motion is for^ 
 ward, viz. from R toward n, and at i where it 
 moves parallel to RM, it is ftationary-, that is, 
 when P is in A. Thro' 1234, R moves backward, 
 or towards V ; and then P is in AZ. 
 
 Cor. 4. In general^ the nodes are always regreffive, 
 except when P is between a node, and its neareft qua- 
 drature \ and then they are progrejfive, wherever the 
 nodes are Jituated. 
 
 Cor. 5. The nodes go fajleji hack when the planet 
 is in T and t. 
 
 For then JR. is at 3 and 7. 
 
 Cor. 6. The incli?iation varies moji, when P is at 
 A and E. 
 
 For then R is at i and 5. 
 
 Cor. 7. And from the various Jttuation of the 
 nodes, and the place of P, it may eajily be determined, 
 'when the inclination increafes or decreafes, in any cafe» 
 
 Cor. 8. Hence if the quantities of thefe forces were 
 known, it would 7iot be dificult to delineate the motion 
 of the pole R, upon the fpberical fur face RXFH ; 
 and at any time to find the inclination, and place of 
 the node. 
 
 Cor. 9 . And to firJ the nature of the curve R 1 2 34 
 defer ibed by the pole R. Suppofing the force directed 
 always to the fun ; and to be as the difiance of P 
 from the plane GZ. 
 
 Let RDB be the cui-ve, and let the tangent /DT 32, 
 revolve about the curve RDB, beginning at R, fo 
 
 F w
 
 66 CENTRIPETAL FORCES. 
 
 Fig. as the end T may move uniformly thro' all the 
 
 32. points of the compafs, in the fame manner as P 
 moves thro' its orbit ATZE/. It is plain this is 
 one property of the curve RDT. 
 
 Now fmce the fun's rays fall at the fame obli- 
 quity upon all parts of the plane ATQ^ therefore 
 the force to draw P in a direction parallel to thefe 
 rays, being the fame at equal diftances from the 
 plane GZ, and always as the diftance •, therefore by 
 the refolution of motion, the diftance that P is 
 drawn perpendicularly from the plane of its orbit, 
 will alfo be as that diftance ; and that is as the va- 
 riation of the orbit's inclination. Therefore if P, 
 inftead of moving to 2 move to 3, then the force 
 at P or PB (fig. 31.) will be as the angle 2P3 : fuppof- 
 
 2 2. ing the fun's diftance from the node to remain the 
 fame, during one revolution of P. 
 
 But when the fun or the force alters its pofition, 
 
 54. it will be greater or lefs on that account, in pro- 
 portion to the fme of OL (where OL is perp. to 
 AL), and that is as the fme of AO, the diftance 
 from the node, the angle A being given. From 
 hence it follows that univerfally, the force afting 
 on P will be always as BP X S.AO; that is, as 
 S.GP X S.AO (fig. 31.); that is, as the fme of 
 the diftance of P from the quadratures, and the 
 fme of the diftance of the fun at O from the node. 
 Now let us find the nature of the curve Ri234^ 
 
 02. fuppofing AO to remain the fame for one revolu- 
 tion of P. Put RA = ;^, AD =: y, RD =z z. 
 Since by the generation of the curve, the angle 
 AD/ zz arch GP, and the force is as the fme of 
 
 X 
 
 GP or of AD/, and — z: S.AD/. Alfo it is plain, 
 the increment of the curve at D is as that force ; 
 
 X 
 
 therefore -^ is as z. And fince in pafting thro' the 
 
 parti*
 
 Sed. III. CENTRIPETAL FORCES. 67 
 
 particle of the curve z<, the line Dt is fuppofed to Fig* 
 change its diredtion uniformly, therefore the angle 32. 
 
 • ^ • 
 
 of contad is given ; whence 2; or -is as the radi- 
 
 z 
 
 r zy , . X . zy 
 
 us 01 curvature, or as -:r ; that is, - is as -^> or 
 
 axx zz z'-y (z being given), and the fluent is 
 ~ zz. yz\ and x : z : : y/jy : ^\a : : as the fub* 
 
 tangent : to the tangent ; which is the property of 
 the cycloid, ^a being zz CB, the diameter of the 
 generating circle. 
 
 Now at different diftances of O from the node, 
 the cycloid defcribed will be greater in proportion 
 to the fine of AO (fig. 31.) ; and even in the fame 
 cycloid, the latter part will be greater than the 
 former part, as AO grows greater ; all the parts 
 of it increafing as the fine of AO increafes -, and 
 the greateft cycloid will be when A is in the qua- 
 dratures ; and the leaft when in the fyziges, where 
 it is reduced to nothing. 
 
 Scholium. 
 
 From the foregoing folution, thefe obfervations 
 may be made. 
 
 J. Tho* the curve R24 has been determined to 
 be a cycloid, yet it is nearer an epicycloid. For 
 at R it fets off nearly in a direftion perp. to GZ, 
 and during its generation (that is, whilft P per- 
 forms a femirevolution) the point A moves to- 
 wards G ; and fuppofmg the force at O to be fix- 
 ed, the lad particle of the curve at 4 would be 
 parallel to that at R. But as O really moves for- 
 ward, fome number of degrees, fuppofe 14, and 
 continues to do fo, all the femirevolution ; there- 
 fore every particle of the curve will have other di- 
 redions in its defcription, being more curve than 
 F 2 before j
 
 68 CENTRIPETAL FORCES. 
 
 Fig. before ; and at laft the tangents at R and 4, will 
 32. make an angle of 14 deg. which is the lame as if 
 an epicycloid was defcribed on the convex fide of 
 a circle, going thro' an arch of it equal to 14 de- 
 grees. 
 
 2. Thus the curve defcribed by R would be 
 nearly an epicycloid, when the force at O is every 
 where of the fame quantity ; yet as O moves about, 
 the force will increafe and decreafe in proportion to 
 the fine of AO ; therefore, if you will fuppofe 
 fuch an epicycloid defcribed as above-mentioned, 
 and moreover imagine the radius of the generating 
 circle to fwell or increafe, in the fame ratio as the 
 S.AO increafes •, then fuch an epicycloid will near- 
 ly reprefent the curve defcribed by R. For then 
 every part of it will be greater or lefs, in propor- 
 tion to the force that generates it. But enough of 
 this., All that I fhall add on this head is the folu- 
 rion of the two following problems, upon account 
 of their curiofity, as depending on the foregoing 
 principles. 
 
 PROP; XXXIII. Prok 
 
 i 
 
 To find the difturbing force of Jupiter or Saturn, 
 upon the earth in its orhit ; having that of the fun 
 upon the moon given. 
 
 2 6. Let the matter in the fun and Jupiter be as m to 
 ^ I. E, I, L the periodic times of the earth, Ju- 
 piter and the moon. A, B, the diftances of the 
 earth and Jupiter from the fun. D the moon's dif- 
 tance from the earth. C, r, the centripetal forces 
 of the fun and Jupiter. 
 
 Then (by Prop. XXVII.) the difturbing force of 
 ■jPK PT 
 
 S upon P, is "Ft^C or as qY^- Therefore if S be 
 
 the fun, and P the moon, the difturbing force 
 
 is 
 
 I
 
 V\.W./^a.{?'d
 
 Onfy^il,;,- 
 
 P1.IV>;(W 
 
 r
 
 Sea. III. CENTRIPETAL FORCES. 69 
 
 D Fig;, 
 
 is -^C i but if S be Jupiter, P the earth, T the ^f. 
 
 fun i then the force is -^ c. That is, the fun*s dif- 
 
 turbing force upon the moon, is to Jupiter's dif- 
 
 D A 
 
 turbing force upon the earth ; as -r- C to ^^ i oras 
 
 DBG to AV. But (Cor. 2. VIL) C 1= -^^ and 
 
 c — -pD* Therefore the fun's force upon the moon, 
 
 DB^ AA 
 
 IS to Jupiter s upon the earth ; as -~rT to -^^j or 
 
 as DB';;/ to A+i that is (Prop. XVIL), as l^Vm : 
 AE'. That is, the fun's difturbing force upon the 
 moon, is to Jupiter's difturbing force upon the 
 earth ; as D X H X m^ to A X EE; But that of 
 the moon is known, and confequently that of Ju- 
 piter. And if for I and ?;?, we put Saturn's peri- 
 odic time, and quantity of matter ; Saturn's dif- 
 turbing force will be known. 
 
 Cor. I . The angular errors generated in the moon 
 by the fun, are to the errors generated in the earth 
 by Jupiter in the fame time^ : : as IlL X w, to E'. 
 
 For (Prop. XXVI H. Cor. 2.) thefe errors are as 
 the forces and periodic times, divided by the dif- 
 tances. Therefore the fun's effecSt to Jupiter's, is as 
 D X n X ;» X L A X E" X E 
 D ^^ A » ^^ ^^ IIL/« to E'». 
 
 Cor. 2 . Hence the error generated in the moon by 
 the fun. is to the error generated in the earth by Jum- 
 pier \ as 1 1230 to I, and to that generated by Sa- 
 turn, as 196076 to I. 
 
 For put I zz 4332 4 days, L = 2 7 |, ;;; zi 1067^ 
 VLm 
 tL ■=. 365-1-, then -p7~ — 11230, And putting 
 
 F 2 I ~
 
 ^o CENTRIPETAL FORCES. 
 
 Fig. I ::= 10759^, and m zz 3021, for Saturn; then 
 26. VLm . ^ 
 
 Cor. 3. Ti;^ /jr<:f cf Saturn to the force of Jupi- 
 ter to dijiurh the earth, is as i to ij ^. 
 
 Cor. 4. 'The motion cf the nodes of the earth's cr- 
 iit by Jupiter's aBion, in 100 years, is ^o' 20" -. And 
 by Saturn's, 35" -|-. 
 
 For the motion of the moon's nodes in a year is 
 19° 20' 32", or 69632"; this divided by 11230 
 gives 6". 2005, multiplied by 100, is 62o".05, which 
 increafed in the ratio of the cofine of inclination of 
 Jupiter's orbit (1= 19' lo"), to that of the moon's 
 (^= 8' -'-), produces 10' 22'' 4^. Which diminifhed in 
 the ratio of i to 17 4, gives 35"^ for Saturn. 
 
 Cor. 5. The motion of the earths aphelion by tht 
 action of Jupiter, is 21' 44" in 100 years, in confe- 
 quentia. And by Saturn, i i^' ^. 
 
 For the motion of the moon's apogee is 40° 40' 
 43", or 14.64.4^" in a year. This divided by 1 1230. 
 gives 13.04"-, which multiplied by 100, gives 
 1304" or 21' 44". And divided by 1 7 4-, gives 74" -4. 
 
 PROP. XXXIV. Prob, 
 
 3 5' To find the variation of inclination of the earth's 
 orbit, by the a5iion of Jupiter in 1 00 years ; and the 
 like for Saturn. 
 
 Let V* 6p t£s yf be the ecliptic, or plane of the 
 earth's orbit •, GFH the orbit of Jupiter -, G Jupi- 
 ter's afcending node ; E, I, Q, the poles of the 
 ecliptic, Jupiter's orbit, and the equator, refpec- 
 tively ; ECk a circle parallel to GF ; and DwQ^a 
 circle parallel to the ecliptic. The pole Q^ here 
 moves regularly along the circle Q/D, by the pre- 
 
 cefTion.
 
 Sea. III. CENTRIPETAL FORCES. 71. 
 
 ceffion of the equinoxes ; wJiich circle is no way Fig, 
 afFededor altered by Jupiter's action; becaufe Ju- 25. 
 piter cannot be fuppoicd to have any force to move 
 the equinoctial points, or alter their regular motion. 
 But he has a power of afting upon the whole body 
 of the earth, and altering its orbit, and confequent- 
 ly the pok E of the ecliptic ; which pole is there- 
 fore made to move along the circle ECK. There- 
 fore we muft fuppofe the orbit of Jupiter fixed, and 
 confequently the pole I, and circle ECK. And 
 now we have to compute the motion of E along 
 the circle ECK. 
 The precefllon of the equinoxes in 
 
 100 years is — — i"" 23' 20" 
 
 And (by the laftprob.) the motion of 
 
 Jupiter's nodes in 100 years is 10' 2 2''-'- or 622 "4- 
 Jupiter's afcending node G (1755, 
 
 angle QEG) — — 69 8» 20' 
 
 Inclination of Jupiter's orbit — i'^ 19' 10". 
 Therefore make the angle QE^ =z i" 23"4-, and 
 EIC zz 10' 2 2 -'-. Upon iLa let fall the perp. Co 5 
 then E<9 is the decreafe of EQ or E^, which 
 (Prop. XXX II.) is the fame as the decreafe of the 
 inclination of the planes of the ecliptic and equi- 
 noctial. 
 
 In the triangle EIC, by reafon of the very fmall 
 angle EIC, \vc Ihall have as rad : S.IC {i" icj 10") 
 
 EIC X S 'C 
 : : angle EIC (622" J„) : EC zi ^^^ = 
 
 i4".3. To the angle GEQ (8^ 20'), add QE^ (i* 
 23' -f ), then OiiQ or <?EC zz 9' 43' \. Then in the 
 very fmall right lined triangle EO, rad : EC : : 
 
 EC X cof oEC 
 cof oEC (9'43'-s) : Eo iz --^ z: 14".!. 
 
 _^ EIC X S.IC X cof. (?RC \ , ^ ^ 
 Or Ed? zz j7 5 the decreafe or 
 
 inclination of the equinoctial in i OQ years by the 
 F 4 actio.a
 
 CENTRIPETAL FORCES. 
 
 aftion of Jupiter. And this decreafe will amount 
 to a minute in 425 years. 
 
 If the fame computation was applied to Saturn, 
 putting EIC ■=. Z5" h I^ =: 2° 30' 10", d?EC n 
 (21° 21' 36" + 1° 23' 4J 22° 44 ^6' ; the decreafe 
 hy Saturn will be i".44. Therefore tlie decreafe 
 by both will be i5".54-, which will be a minute in 
 386 years. 
 
 Cor. I . 'The inclination will decreafe till E and a 
 he at their neareft dijiance in the two circles^ which, 
 ivill he above 6000 years ; and then it will increafe 
 again. It haslikewife heen decreafing for above 8000 
 years. 
 
 For the diameters of the circles EK and DQ, 
 being about as i to 1 7. And the angle lEQ being 
 81"' 40', and the difference of the motions of E 
 and Q being i ° 13'; it will decreafe nearly as ma- 
 ny centuries as is the quotient of 8 1 ° 40' divided 
 by 1° 13', which is ^^. Alfo the fupplcmcnt 98° 20' 
 divided by 1° 13', gives 81 centuries, it has been 
 decreafing. 
 
 Cor. 2. But the increafe or decreafe for every cen- 
 tury is not 1 5''. 54, as determined in this particular Ji-^ 
 tuation. For as it approaches to its maximum or mini' 
 mum., it varies very flow ^ and at thefe places is at a 
 fiand for a long time. 
 
 Cor. 3. The inclination can never he lefs than ^0° 
 50' 54"-, nor greater than 16° 7' 20". 
 
 For the neareft and greateft diftances of the two 
 circles EK, DQ^amount but to thefe. And there 
 muft be many revolutions, before they can light 
 upon thefe tv/o points, if the world can be fup- 
 pofed to exift fo long. 
 
 Scholium. 
 
 The difturbing forces of Jupiter and Saturn here 
 made ufe of are derived from that of the fun up- 
 on
 
 Sea. III. CENTRIPETAL FORCES. 73 
 
 on the moon -, but thefe forces are really more than Fig. 
 are here determined. For in calculating the dif- 35, 
 turbing force of the fun (by Prop. XXVII.), the 
 forces upon T and p are as rr to rr + iry -\- yy\ 
 but by reafon of the great diftance of the fun, the 
 part yy is left out, as being extremely fmall. But 
 in the cafe of Jupiter and Saturn it is otherwife, 
 and therefore yy mud be taken in. Whence the 
 difturbing force muft have an additional increafe, 
 which is as 2.ry to yy^ or as ir tojy, which in Jupi- 
 ter is as 10 to I, and in Saturn as 19 to i. There- 
 fore Jupiter's difturbing force muft be increafed by 
 ^Vh, and Saturn's by -'-^th ; which being done, 
 their effefts will be proportional, and the decreafe 
 of inclination of the ecliptic in 100 years, by Ju- 
 piter and Saturn will become i5''.45, and i".5i ; 
 and by both i6".96 or 17" nearly; which will 
 mount to a minute in 353 years. 
 
 But if the obfervations of the antients can be 
 depended on, the obliquity decreafes fafter than this. 
 For by the obfervations of Arifiarchus^ Eratojihenes^ 
 Hyfcirchus, Ptolemy^ and Theon^ the obliquity was 
 found to be 23^ 51'. And none of them lived 
 300 years before Chrift, and two of them after. 
 So that in little m.ore than 2 coo years, there is a 
 difference of 22'; which is more than a minute in 
 100 years, and is m.ore than three times as faft as 
 we have here determined it. 
 
 I ftiall now proceed to fome things of another 
 kind, relating to centripetal forces ; which as far 
 as I can find, have not been m.eddled with by any 
 body before. 
 
 PROP. XXXV. Proh. 
 If the circle GDFE he moved along the right line 2$, 
 AB, whilji it turns round its axis i to find its motion 
 upon a horizontal plane. 
 
 Suppofe the circle inlined in any given angle to 
 the horizon, the line of direction AB being at firfl: 
 
 in
 
 74 CENTRIPETAL FORCES. 
 
 Fig. in its plane ; and let it move round its axis CP, 
 56. which is perpendicular to its plane, with any velo- 
 city. Let C5 be the center of ofcillation. 
 
 Now the circle will endeavour to defcend by its 
 gravity, in the fame manner, as the fingie point O 
 would do. Therefore fuppofe the point F to de- 
 fcend thro' F/ in a moment of time, and that F 
 is transferred to I in the fame time. Then if the 
 parallelcgram F/«I be compleated, the point F 
 by the compound motion, v/ill move along Fn. 
 By this means CP, the axis of revolution, will be 
 transferred to the pofition C^, inclining more to- 
 wards B. But when the circle has made half a 
 revolution, and G is come to the place F -, the 
 points in F, proceeding in the tradt Fn, will move 
 the axis C^ forwards, as before; that is (by the 
 turning of the circle) it will throw it into its former 
 place CP. So that during a revolution, the axis is 
 thrown contrary ways in all the oppofite points, 
 and fo is always reftored back to its firfl place in 
 regard to the plane. Therefore the circle always 
 revolves about the axisCP, whilft CP continually 
 inclines more and more forward •, that is, the plane 
 of the circle continually alters its pofition ; and 
 the variation of its pofition is known from the lines 
 FI, nl -, and is equal to the angle nFl or FCq. 
 And fince the circle endeavours to move along a 
 line which is in the plane «FG, it will no longer 
 go along AB, but deviate from it into a new tradl, 
 which is now to be found. 
 
 It may be convenient to imagine the circle to 
 be a poligon with an infinite number of fides. 
 Then let KL be the horizon, M^ the plane of the 
 the circle, g being a point in the right line AB ; 
 let the next point of the circle (or angle of the po- 
 ligon) defcend to the horizon, thro' the very fmall 
 fpace r/-, then in the right angled triangle irg, S . 
 inclination (t^r) : /r or nl ("which is as S . «FI) : :
 
 JtZ.. J4-
 
 tttt^tt;
 
 Sea. III. CENTRIPETAL FORCES. y^ 
 
 S.nFl ^, r -r 1 . . » Fife, 
 rad ( I ) : tg = g ' Therefore if the circle be |; 
 
 moved thro' Gg in the fame time, then from g (in 
 
 Sn¥l 
 the hne AB ; fettino; ongt — ■^. — r- — ~. — , then t 
 y G «b S.inchnation' 
 
 will be a point in the curve G/R, thro' which the 
 
 circle will pafs. After the fame manner it will 
 
 again deviate from the laft diredion tlf ; and de- 
 
 fcribe the curve GRVWX. 
 
 Generally the whirling motion round its axis, is 
 
 equal to its progreflive motion ; for the friftion of 
 
 the plane foon reduces it to that. But take away 
 
 the friftion, and thefe two different motions may 
 
 be v/hat you will. 
 
 Cor. I. The curvature in any j>lace, is. reciprocally 
 as thefe three quantities^ the velocity of rotation^ the 
 frogreffive velocity^ and the tangent of inclination. 
 
 For the curvature is as the angle tQg^ that is, as 
 
 tg S.«FI 
 
 7=:- or as .-, , c- • — T '- — » that is, as 
 
 M? G^X S.inciinaLion ' 
 
 »I . Cof. incl. 
 
 FlxG^XS-inclin". ' ^^^^^ '^' ^^ FI X G^ X S.inclin. 
 I 
 
 or as Tjt V <- ' v. ^ '■ — V — ' For n\ is as the co- 
 
 Jb 1 X C:j^ X tan. mclin. 
 
 fine of inclination, being the fpace defcribed upon 
 the inclined plane 7i\. 
 
 Cor. 2 . Taking away all iinfediments^ the circle al- 
 ways keeps the fame inclination to the horizon. 
 
 For the pofition of the plane F«I is fuch, that 
 the axes CF, Cj, are both parallel to it. If we 
 fuppofe gravity to a6l by a fingle impulfe at O, 
 then F will move to n., and P to ^. And the plane 
 of the circle endeavouring to defcend a little at D, 
 and rife at F ; a new point of the circle as /, ly- , 
 ing beyond G will inftantly touch the plane ; by 
 which means it leaves the line AG. And fince at 
 
 every
 
 CENTRIPETAL FORCES. 
 
 . every point of contaft as G, the pole P moves (at 
 each impulfe of gravity) in a line perp. to CG, 
 and alfo parallel to the tangent arch at G ; and the 
 like at every new point of contad ; it is plain CP 
 is always alike inclined to the horizon Confe- 
 quently, when gravity is continual, the circle 
 coming continually to new points of contad, the 
 axis CP will always revolve round at the fame in- 
 clination, and therefore the plane of the circle will 
 alfo have the fame inclination to the horizon. 
 
 This might alfo be proved after the manner of 
 the XXXth Prop, not confidering the progreflive 
 motion of the circle. 
 
 All this is fuppofmg there is no refiftance, fric- 
 tion, or other irregularity. But fince in fa6l, the 
 refiftance of the air continually leflens its motion, 
 and the fmoothnefs of the plane it runs on, caufes 
 the foot or bottom of the circle to Aide outward, 
 which continually leflens the inclination, and brings 
 the axis more upright ; and the more oblique the 
 plane of the circle is, the fafter it Aides out. Up- 
 on thefe accounts it can never defcribe a circle, 
 but only a fort of fpiral line ; and the plane of the 
 circle defcending lower and lower, at lad falls fiat 
 ■upon the horizon. 
 
 Cor. 3. Hence a circle moving without any refif- 
 tance^ i^c. upon a horizontal plane ; tcvV/ defcribe a 
 circle upon that plane. 
 
 For the velocity and inclination continuing the 
 fame •, the curvature of the tradt defcribed, will be 
 every where alike. 
 
 Cor. 4. And to find the diameter of the circle or 
 orbit defcribed. 
 
 Let / be a very fmall part of time wherein G^ is 
 defcribed, v the velocity of projeftion per fecond, 
 h the fpace defcended by gravity in time /, s and c 
 the fme and cofine of the circles inclination, / ~ 
 
 16
 
 Seam. CENTRIPETAL FORCES. 77 
 
 1 6 -i-V feet -, then will ch zz fpace defcended along Fig. 
 the inclined plane F/j and by the laws of motion, 36. 
 1" : 1; : : t" : Gg - tv. 
 
 h 
 zndit : h ::i": f = --• 
 
 Then whilft O has moved thro' the length Gg, t or 
 I has defcended thro* the fpace ch on an inclined 
 
 n\ 
 
 plane parallel to F/. But we proved gt zz. ~ zn 
 
 tr ch 
 
 -7 n — • And therefore the diameter of the orbit 
 s s 
 
 ~ If " Ih ~" ch "^ cf — f 
 cli nation. 
 
 Cor. 5. Alfo to find the periodic time, or time of 
 ine revolution. 
 
 Let D iz diameter of the orbit, then by Cor. 
 
 G? X s ^^^ 
 
 laft, lH-pi = D, and Gg = s/'-T' The cir- 
 ch •* 
 
 cumference of the orbit is ^ (putting tt 
 
 ~ 3.1416^; and by uniform motion, Gg -. t : ; 
 
 ^ : T the periodic time zz. —j- 
 
 ch^ ttsD SD 
 
 Or thus, 
 G^ (tv) : t : : circumference -y — : periodic time 
 
 — -7- = "F X tan. inclination. 
 
 PROP,
 
 78 CENTRIPETAL FORCES. 
 
 ^^' PRO P. XXXVI. Prol^. 
 
 ^7. ^f DEF h the furface of a right cone^ ijohofe 
 mis AE is -perpendicular to the horizofi^ and DHFG 
 a circular plane parallel to the horizon ; and if a cir- 
 cle ah revolves round in the periphery DHFG, with 
 its axis i^e always parallel to the fide of the cone 
 DE, where it then is. To find the periodic time of 
 ah, in the circle DHFG. 
 
 Draw DBA perp. to DE, and BC perp. to AC. 
 Let TT zz 3. 141 6, h — fpace defcendcd by gravity 
 in the time t. 
 
 Then if the force of gravity be reprefented by 
 AC, the centrifugal force at B to keep the circle 
 ah in that pofition, thro' its whole revolution, will 
 be denoted by BC. Then it will be AC (the gra- 
 
 BC 
 vity) : BC (the cent, force) : -. h -, -r^ h — fpace 
 
 defcended towards C, by the force in direction 
 BC — the effedl of the centrifugal force. There- 
 fore by (Prop. II.) the periodic time is tt/ y/ 
 2BC ^2AC 
 
 Cor. I. Hence the periodic ti-me zn 7rt 
 
 J 
 
 —J— X tan. inclination ABC. 
 
 S.B ^ S.B 
 
 For S.A : S.B : : BC : AC =: 3;^ X BC = -^g 
 
 X BC zz tan. B X BC ; whence, the periodic time 
 
 /2BC ~7ZZ 
 ~ 7!" / ^ — - X tan. ABC. 
 
 Cor. 2. Draw BI parallel to DE, and DL pa-^ 
 
 7r/ X BC X v/2 
 
 ralkl to?,Q-, then the periodic time iz — .. j , * 
 
 T. . ^ BC^ 
 
 For AC = ^j- . C^^^
 
 J 
 
 Sea. III. CENTRIPETAL FORCES. 79 
 Cor. 3. Hence the curve defcribed on the conic fur- Fig. 
 face, is the fame as that defcribed on a horizontal 37. 
 plane, as explained in the lafi Prop. All the difference 
 is, that the conic fiirface hinders the circle ab from 
 Jliding outzcard, which the horizontal plane cannot do ; 
 except it be. fuppofed to he fo rough, that the circle can- 
 not Jlide on it. 
 
 For in Cor. 4. of the laft Prop, v is put for the 3^* 
 velocity of G along AB •, but putting it for the 
 velocity at C, you'll have the diameter of the orbit 
 paffing thro' C : that is, (fig. 37.) inftead of 2DL 
 we Ihould find 2BC for the diameter D of the orbit. 37' 
 
 And by Cor. 5, the periodic time is tt ^ \/t~- ^^^ 
 in Cor. i. of this Prop, the periodic time is -rrt 
 
 2BC . \ ; 
 
 h ^ ^^^' ^^^^^^^^^^^ i which is equal to the 
 
 former, becaufe D — 2BC, and — — tan. of the 
 ' c 
 
 inclination. So the curve is the lame in both cafes. 
 
 PROP. XXXVII. Proh. 
 
 I'o explain the motion of a top, or fuch like whirl- 38. 
 ing body 
 
 Let ABC be a top w^hirling about in the order 
 AEC •, FDG a circle defcribed by any point D in 
 the furface, K its center of gravity, BKca the axis 
 of the top. 
 
 \i the top be upright upon the foot B ; that Is, 
 if BS be perp. to the horizon, and moves fwiftly 
 about ; it will continue upright till the motion 
 flacken. But when it is going to fall, it will lean 
 to one fide ; therefore fuppofe D to be the loweft 
 point in the circle FG ; then the top endeavours by 
 its gravity t(j defcend towards D. Let the force of 
 gravity a^one move the point D thro' the fpace D<7, 
 in a very fmall time y during which, the rotary 
 
 motion
 
 8o CENTRIPETAL FORCES. 
 
 Fig. motion would carry the point D to r. Compleat 
 38. the parallelogram Drpo, and the point D will be 
 carried thro' Dp ; that is, the circle FDG will come 
 into the pofition Dp •, and therefore the axis BKS 
 (perp. to the circle FG) v/ill be moved in a direc- 
 tion towards H, perp. to DK -, and the point S 
 moved to n ; the particle Sn being parallel to Dp. 
 After the fame manner by a new impulfe of gravi- 
 ty at/, the loweft point-, the circle FDG, will be 
 moved into a new pofition, below Dp, and the 
 point S carried from n to t. And fo by every im- 
 pulfe of gravity, the point S will be moved gra- 
 dually forward, thro' the circle Sntalz •, and thus 
 the top recovers itfelf from falling ; the motion of 
 S being always parallel to that of D. And tliere- 
 fore the motion of the axis BS will be the fame 
 way about, as the top's motion is. And thus the 
 point S will continue to make feveral revolutions 
 by a flow motion, whilft the top makes its revolu- 
 tions about its axis, by a fwift motion. 
 
 Cor. I. J'his motion cf the top and, its axis, is fi~ 
 milar to the motion of an oblong fpheroid, and its 
 nodes. 
 
 For CCor. 6. Prop. XXX.) the nodes move the 
 fame way about as the body revolves, and fo does 
 the axis of the top •, and therefore this motion may 
 be called the motion of the nodes of the top. 
 
 Cor. 2. When the tops tnotion is very pwift^ the 
 circle Sql is very fmall •, and as it grows /lower, that 
 circle will grow bigger and bigger, till the top falls. 
 
 For when the top's motion is very fwift, Dr will 
 be greater, and the angle rDp lefs -, and the circle 
 Dp will deviate lefs from DG. And gravity hav- 
 ing little power to difturb its motion, the circle Sj/ 
 will be extremely fmall, and the top will revolve 
 about the axis in appearance unmoved. But as the 
 
 top's
 
 P1.VI.//^Z////
 
 w 
 
 Kg-. ;((i 
 
 r 
 
 i ' 
 
 37- 
 
 K 
 
 , y. 
 
 G 
 
 
 
 
 
 if=-£" 
 
 L 
 
 ! . 
 
 ^ 
 
 ^\3il- 
 
 n
 
 Sea. III. CENTRIPETAL FORCES. St 
 
 top's motion by refiftance and fridion grows lefs Fio-. 
 and lefs, Dr will be lefs ; and the circle Dp will de- o 8° 
 viate more from DG ; that is, gravity will have 
 more and more power to difturb its motion ; and the 
 axis BS will defcribe a greater and greater circle 
 with the point S, or rather a fpiral, till at laft the 
 top falls down. 
 
 Cor. 3. y^s the top grows flow, and the motion 3^' 
 ijoeak, and the pole S defcribes greater and greater cir^ 
 cles, the foot B is thrown out to the oppofite ftde, de-^ 
 fcrihing a circle ^b, which is greater as Sql is greater ; 
 and goes the fame way about. 
 
 For the center of gravity K always endeavours 
 to be at reft, whilft the body revolves about. 
 Therefore when the top grows weak, and the pole 
 S defcribes greater circles, the foot B is thrown fur- 
 ther out to the oppofite fide ; and being always op- 
 pofite, will defcribe a circle proportional to Sj/j 
 the foot B going the fame way about as S does. 
 And thefe circles B^ will continually grow greater 
 and greater till the top falls down. Till then the 
 top rolls about and about from the pofition CAB 
 to the oppofite pofition cab, till the motion end, 
 and the top falls down. And thefe are the princi- 
 pal phcenomena of the motion of a top. 
 
 F I N I ^.
 
 Page 
 I 
 
 Lin 
 II 
 
 59 
 
 19 
 
 6S 
 
 I 
 
 78 
 
 6 
 
 ERRATA, 
 read 
 (Preface) irregularities only 
 PtiC and Pre. 
 4, R recedes from M, 
 its axis I Bz 
 
 In the Plates. 
 
 Fig. 5. m fhould be Ihaded. 
 
 Jig. 23. P« ftjould be perp, to F/,
 
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