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LEMMA I. Quantities proportional to their differences are continually pro

portional. Let A be to A - B as B to B - Cand C to CD, &c. and, by conversion, A will be to B as B to C and C to D, &c. Q.E.D.

PROPOSITION II. THEOREM II. If a body is refifted in the ratio of its velocity, and moves, by

its vis infita only, through a similar medium, and the times be taken equal, the velocities in the beginning of each of the times are in a geometrical progression, and the spaces described in each of the times are as the velocities. (Pl.'1, Fig. 1.)

Case 1. Let the time be divided into equal particles; and if at the very beginning of each particle we suppose the refiftance to act with one single impulse which is as the velocity, the decrement of the velocity in each of the particles of time will be as the same velocity. Therefore the velocities are proportional to their differences, and therefore (by lem. 1, book 2) continually proportional. Therefore if out of an equal number of particles there be compounded any equal portions of time, the velocities at the beginning of those times will be as terms in a continued progreflion, which are taken by intervals, omitting every where an equal number of intermediate terms. But the ratios of these terms are compounded of the equal ratios of the intermediate terms equally repeated, and therefore are equal. Therefore the velocities, being proportional to those terms, are in geometrical progreffion. Let those equal particles of time be diminished, and their number increased in infinitum, so that the impulse of resistance may become continual; and the velocities at the beginnings of equal times, always continually proportional, will be also in this case continually proportional. Q.E.D.

Case 2. And, by division, the differences of the velocities, that is, the parts of the velocities lost in each of the times, are as the wholes; but the spaces described in each of the times are as the loft parts of the velocities (by prop. 1, book 2), and therefore are also as the wholes. Q.E.D.

Corol. Hence if to the rectangular asymptotes AC, CH, the hyperbola BG is described, and AB, DG be drawn perpendicular to the asymptote AC, and both the velocity of the body, and the resistance of the medium, at the very beginning of the motion, be expressed by any given line AC, and, after fome time is elapsed, by the indefinite line DC; the time may be expressed by the area ABGD, and the space described in that time by the line AD. For if that area, by the motion of the point D, be uniformly increased in the same manner as the time, the right line DC will decrease in a geometrical ratio in the same manner as the velocity; and the parts of the right line AC, described in equal times, will decrease in the fame ratio.

PROPOSITION III. PROBLEM I. To define the motion of a body which, in a similar medium,

afcends or descends in a right line, and is refifted in the ratio of its velocity, and acted upon by an uniform force of gravity. (Pl. 1, Fig. 2.)

The body ascending, let the gravity be expounded by any given rectangle BACH; and the resistance of the medium, at the beginning of the afcent, by the rectangle BADE, taken on the contrary side of the right line AB. Through the point B, with the rectangular asymptotes AC, CH, describe an hyperbola, cutting the perpendiculars DE, de, in G, g; and the body ascending will in the time DGgd describe the space EGge; in the time DGBA, the space of the whole ascent EGB; in the time ABKI, the space of defcent BFK; and in the time Ikki the space of descent KFfk; and the velocities of the bodies (proportional to the resistance of the medium) in these periods of time will be ABED, ABed, O, ABFI, ABfi respectively; and the greatest velocity which the body can acquire by descending will be BACH.

For let the rectangle BACH (Pl. 1, Fig. 3) be resolved into innumerable rectangles Ak, Kl, Lm, Mn, &c. which shall be as the increments of the velocities produced in so many equal times; then will 0, Ak, Al, Am, An, &c. be as the whole velocities, and therefore (by fuppofition) as the refiftances of the medium in the beginning of each of the equal

times. Make AC to AK, or ABHC to ABKK, as the force of gravity to the resistance in the beginning of the second time; then from the force of gravity subduct the resistances, and ABHC, KKHC, LIHC, MmHC, &c. will be as the absolute forces with which the body is acted upon in the beginning of each of the times, and therefore (by law 2) as the increments of the velocities, that is, as the rectangles Ak, Kl, Lm, Mn, &c. and therefore (by lem. 1, book 2) in a geometrical progression. Therefore, if the right lines Kk, L1, Mm, Nn, &c. are produced so as to meet the hyperbola in q, r, s, t, &c. the areas ABqK, KqrL, LrsM, MstN, &c. will be equal, and therefore analogous to the equal times and equal gravitating forces. But the area ABqK (by corol. 3, lem. 7 and 8, book 1) is to the area Bkq as Kq to fkq, or AC to 4AK, that is, as the force of gravity to the resistance in the middle of the first time. And by the like reasoning the areas qKLr, rLMs, sMNt, &c. are to the areas qklr, rlms, smnt, &c. as the gravitating forces to the resistances in the middle of the second, third, fourth time, and so on. Therefore fince the equal areas BAKq, qkLr, rLMs, sMNt, &c. are analogous to the gravitating forces, the areas Bkq, qklr, rlms, sinnt, &c. will be analogous to the resistances in the middle of each of the times, that is (by supposition), to the velocities, and fo to the spaces described. Take the sums of the analogous quantities, and the areas Bkq, Blr, Bms, Bnt, &c. will be analogous to the whole fpaces described; and also the areas ABqK, ABrL, ABsM, ABEN, &c. to the times. Therefore the body, in defcending, will in any time ABrL describe the space Blr, and in the time LrtN the space rlnt. Q.E.D. And the like demonftration holds in afcending motion.

Corol. 1. Therefore the greatest velocity that the body can acquire by falling is to the velocity acquired in any given time as the given force of gravity which perpetually acts upon it to the resifting force which opposes it at the end of that time.

COROL. 2. But the time being augmented in an arithmetical progression, the sum of that greatest velocity and the velocity in the ascent, and also their difference in the defcent, decreases in a geometrical progression.

COROL. 3. Also the differences of the spaces, which are described in equal differences of the times, decrease in the fame geometrical progression.

COROL. 4. Tine fpace described by the body is the difference of two spaces, whereof one is as the time taken from the beginning of the descent, and the other as the velocity ; which [spaces] also at the beginning of the descent are equal among themselves.

PROPOSITION IV. PROBLEM II. Supposing the force of gravity in any fimilar medium to be uniform, and to tend perpendicularly to the plane of the horizon ; to define the motion of a projectile therein, which suffers resistance proportional to its velocity. (Pl. 1, Fig. 4.)

Let the projectile go from any place D in the direction of any right line DP, and let its velocity at the beginning of the motion be expounded by the length DP. From the point P let fall the perpendicular PC on the horizontal line DC, and cut DC in A, fo that DA may be to AC as the resistance of the medium arising from the motion upwards at the beginning to the force of gravity; or (which comes to the fame) fo that the rectangle under DA and DP may be to that under AC and CP as the whole resistance at the beginning of the motion to the force of gravity. With the asymptotes DC, CP describe any hyperbola GTBS cutting the perpendiculars DG, AB in G and B; complete the parallelogram DGKC, and let its fide GK cut AB in Q. Take a line N in the same ratio to QB as DC is in to CP; and from any point R of the right line DC erect RT perpendicular to it, meeting the hyperbola in T, and the right lines EH, GK, DP in I, t, and

IGT V; in that perpendicular take Vr equal to Nor, which is

GTIE the same thing, take Rr equal to

N

; and the projectile in the time DRTG will arrive at the point r', describing the curve line DraF, the locus of the point r; thence it will come to its greatest height a in the perpendicular AB; and after

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wards ever approach to the asymptote PC. And its velocity in

any point r will be as the tangent rL to the curve. Q.E.I. For N is to QB as DC to CP or DR to RV, and therefore

DR X QB RV is equal to

and Rr (that is, RV – Vr, or

N
DR X QB
— UGT) is equal to
DR X AB - RDGT

-. Now N

N let the time be expounded by the area RDGT and (by laws, cor. 2,) distinguish the motion of the body into two others, one of ascent, the other lateral. And since the resistance is as the motion, let that also be distinguished into two parts proportional and contrary to the parts of the motion: and therefore the length described by the lateral motion will be (by prop. 2, book 2) as the line DR, and the height (by prop. 3, book 2) as the area DR X AB - RDGT, that is, as the line Rr. But in the very beginning of the motion the area RDGT is equal to the rectangle DR X AQ, and therefore that line Rr

DR X AB - DR X A (or

N AQ or QB to N, that is, as CP to DC; and therefore as the motion upwards to the motion lengthwise at the beginning. Since, therefore, Rr is always as the height, and DR always as the length, and Rr is to DR at the beginning as the height to the length, it follows, that Rr is always to DR as the height to the length; and therefore that the body will move in the line DraF, which is the locus of the point r. Q.E.D.

DR X AB RDGT Cor. 1. Therefore Rr is equal to

N

N; and therefore if RT be produced to X so that RX

DR X AB equal to

that is, if the parallelogram ACPY be

N completed, and DY cutting CP in Z be drawn, and RT be

RDGT produced tillit meets DY in X; Xr will be equal to and

N therefore proportional to the time.

Cor. 2. Whence if innumerable lines CR, or, which is the same, innumerable lines ZX, be taken in a geometrical

AQ) will then be to DR as AB —

may be

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