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LEMMA X.

The spaces, which a body describes, by any finite force urging it,

whether that force be determined and immutable, or be continually increased or continually diminished, are, in the very beginning of the motion, in a duplicate ratio of the times.

Let the times be represented by the right lines AD and AE, sce figure to the preceding lemma, and the velocities generated, by the ordinates DB and EC ; the spaces described by these velocities, are as the areas ABD and ACE described by these ordinates, or, which is equal by the precediug lemma, in a duplicate ratio of the times AD and AE.

LEMMA XI.

The evanescent subtense of the angle of contact, in all curves hav

ing a finite curvature at the point of contact, is ultimately in a duplicate ratio of the subtense of the conterminous arch.

Case 1. Let that arch be AB, its tangent A AD, the subtense of the angle of contact BD,

16 and the subtense or chord of the contermi

B nous arch or the arch having the same ex- C tremes, the right line AB; and first, let the subtense BD of the angle of contact be perpendicular to the tangent AD. To the subtense AB and tangent AD, erect the perpendiculars BG and AG, meeting each other in G, and let the points D, B and G approach J to the points d, b and g, and let j be the intersection of the right lines BG and AG,

G made ultimately, when the points D and B come to A. It is manifest that the distance GJ may be less than any given right line. But, because of the right angled triangle ABG, the square of AB is equal to the rectangle GXC (Cor. 1. 8 6 and 17. 6 Eu.), or, AC and BD being equal, to the rectangle under AG and BD; for the same reason, the square of the right line Ab is equal to the rectangle under Ag and bd; therefore the square of AB is to the square of Ab, in a ratio compounded of the ratios of AG to Ag and of BD to bd (23. 6 Eu). But because GJ may be assumed less than any given length, the ratio of AG to Ag may be such, as to differ from the ratio of equality less than by any given difference, and therefore the ratio of the square of AB to the square of Ab may be such, as to differ from the ratio of BD to bd less than by any given difference; and therefore, by lemma 1, the ultimate ratio of the square of AB to the square of Ab is equal to the ultimate ratio of bD to Bd, and so bD is to bd ultimately in a duplicate ratio of the subtenses AB and Ab (20. 6 Eu).

Case 2. Let now BD be inclined to AD'in any given angle, and the ultimate ratio of BD to bd will still be the same as before (4. 6 Eu.), and therefore in a duplicate ratio of the subtenses AB and Ab.

Cnse 3. And though the angle D should not be given, but be formed by the right line BD converging to a given point, or by any other law; yet the angles D and d, constituted by a common law, always tend to equality, and approach nearer to each other, than by any given difference, and are therefore ultimately equal, by lemma 1 ; and therefore the lines BD and bd are to each other, in the same ratio as before.

Cor. 1. Since the tangents AD and Ad, the arches AB and Ab, and their sines BC and bc, become ultimately equal to the chords AB and Ab; their squares are ultimately, as the subtenses BD and bd.

Cor. 2. Their squares are also ultimately as the sagittás or versed sines of the arches, bisecting the chords, and tending to a given point. For these sagittas are as the subtenses BD and bd.

Cor. 3. And therefore the sagitta is in a duplicate ratio of the time, in which a body with a given velocity describes an arch; the arch described with a given velocity being as the time.

Cor. 4. The rectilineal triangles ADB and Adb are ultimately in a triplicate ratio of the sides AD and Ad, and in a sesquiplicate of the sides DB and db, as being in a ratio, compounded of the ratios of the sides AD and DB to Ad and db (Cor. 1. 23. 6 Eu). So also the rectilineal triangles ABC and Abc are to each other ultimately in a triplicate ratio of the sides BC and bc (See Def. 14. 5 Eu).

Cor. 5. And because DB and db are ultimately parallel, and in a duplicate ratio of AD and Ad, and therefore AC and Ac ultimately in a duplicate ratio of BC and bc, which is the nature of the parabola (Cor. 2. 40. 1 Sup.); the ultimate curvilineal areas ADB and Adb are two thirds parts of the rectilineal triangles ADB and Adb (Cor. 81. 1 Sup.); and therefore the ultimate segments AB and Ab third parts of the same triangles. And therefore these areas and segments are in a triplicate ratio both of the tangents AD and Ad, and of the chords of the arches AB and Ab.

In subsequent citations, Nat. Ph. denotes, Natural Philosophy.

PROP. I. THEOR.

The areas, which revolving bodies describe, by radiuses drawn to an

immoveable centre of force, are in immoveable plains, and proportional to the times.

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Let the time be divided into equal parts, and, in

E the first part of time, let

a

1 the body describe, by its innate force, the right

HL line AB. The would, in the second part of time, if nothing hindred, go on directly to c, describing the right line Bc equal to AB (by law

KYB 1), so that the radiuses

G. A'S, BS and cs being drawn to the centre S, there would be described equal areas ASB and

А BSc (38. 1 Eu.): but, when the body comes to B, let a centripetal force act with a single but great impulse, and cause that the body deviate from the right line Bc, and go on in the right line BC. Through c draw cC parallel to BS, meeting BC in C; and, the second part of time being completed, the body (by cor. to the laws), will be found in C, in the same plain with the triangle ASB : join SC, and the triangle SBC, because of the parallels SB and Cc, is equal to the triangle SBc (37. 1 Eu.), and therefore to the triangle SAB. By a like reasoning if the centripetal force act successively in C, D, E, &c. causing that the body, in the several particles of time, describe the several right lines CD, DE, EF, &c. all these will be in the same plain, and the triangle SCD will be equal to the triangle SBC, the triangle SDE to SCD, and SEF to SDE; therefore equal areas are described in equal times in an immoveable plain, therefore any sums of the areas SADS and SAFS are to each other, as the times of their alescription (Theor. 2. 15. 5 Eu). Let now the number of

these triangles be increased, and their breadth diminished in infinitum, and their ultimate perimeter ADF (by cor. 4 lem. 3), will be a curve line, as must be the case, since the centripetal force, by which the body is perpetually drawn from the tangent, is supposed to act unceasingly, and any described areas SADS and SAFS, which have been shewn to be always to each other, as the times of their description, are, in this case also, to each other, as the times of their description.

Cor. 1. The velocity of a body, attracted towards an immoveable centre, in non-resisting spaces, is inversely as the perpendicular let fall from that centre, on a rectilineal tangent of the orbit. For the velocity in the places A, B, C, D and E, are as the bases of equal triangles, namely, AB, BC, CD, DE and EF, and these bases are reciprocally as the perpendiculars let fall on them, as is manifest from 15. 6 Eu.

Cor. 2. If the chords AB and BC, of two arches successively described in equal times, in non-resisting spaces, by the same body, be completed into a parallelogram ABCG, and its diagonal BG, in that position, which it has ultimately when these arches are diminished in infinitum, be produced both ways; it will pass through the centre of force.

Cor. 3. If the chords AB and BC, DE and EF, of arches described in equal times in non-resisting spaces be completed into parallelograms ABCG and DEFH; the forces in B and E are to each other in the ultimate ratio of the diagonals BG and EH, when these arches are diminished in infinitum. For the motions of the body BC and EF (by cor. to the lau's), are compounded of the motions Bc and BG, Ef and EH; and BG and EH, equal to Cc and Ff, in the demonstration of this proposition, were generated from the impulses of the centripetal force in B and E, and are therefore proportional to these impulses.

Cor. 4. The forces, with which, any bodies, in non-resisting spaces, are drawn from rectilineal motions, and turned into curvilineal orbits, are to each other, as those sagittas of arches, described in equal times, which tend to the centre of force, and bisect the chords, when these arches are diminished in infinitum. For the sagittas BK and EL, when these arches are so diminished, are balves of the diagonals, mentioned in the preceding corollary (Schol. 3. 13. 2 Eu).

Cor. 5. And therefore, the same forces, are to the force of gravity, as these sagittas, are to sagittas, perpendicular to the horizon, of the parabolick arches, which projectiles describe in the same time.

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