the number of intervals between them and the given term. Let A, B, C, D, E, F, be continually proportional; then if the term C is given, the moments of the reft of the terms will be among themselves, as-2A, -B, D, 2 E, 3 F.

COR. 2. And if in four proportionals the two means are given, the moments of the extremes will be as thofe extremes. The fame is to be understood of the fides of any given rectangle.

COR. 3. And if the fum or difference of two fquares is given, the moments of the fides will be reciprocally as the fides.

SCHOLIUM.

In a letter of mine to Mr. J. Collins, dated December 10. 1672. having described a method of Tangents, which I fufpected to be the fame with Slufius's method, which at that time was not made publick; I fubjoined these words; This is one particular, or rather a corollary, of a general method, which extends itself, without any troublesome calculation, not only to the drawing of Tangents to any Curve lines, whether Geometrical or Mechanical, or any how respecting right lines or other Curves, but also to the refolving other abstruser kinds of Problems about the crookedness, areas, lengths, centres of gravity of Curves, &c. nor is it (as Hudden's method de Maximis & Minimis) limited to equations which are free from furd quantities. This method I have interwoven with that other of working in equations, by reducing them to infinite feries. So far that letter. And thefe laft words relate to a Treatife I compofed on that fubject in the year 1671. The foundation of that general method is contained in the preceding Lemma.

PROPOSITION VIII.

THEOREM VI.

If a body in an uniform medium, being uniformly acted upon by the force of gravity, afcends or defcends in a right line; and the whole space defcribed be diftinguished into equal parts, and in the beginning of each of the parts, (by adding or fubducting the refifting force of the medium to or from the force of gravity, when the body afcends or defcends) you collect the abfolute forces; I fay that thofe abfolute forces are in a geometrical progreffion. Pl. 2. Fig. 1.

For let the force of gravity be expounded by the given line AC; the force of resistance by the indefinite line AK, the abfolute force in the defcent of the body, by the difference KC; the velocity of the body by a line AP, which fhall be a mean proportional between AK and AC, and therefore in a fubduplicate ratio of the refiftance; the increment of the refiftance made in a given particle of time by the lineola KL, and the contemporaneous increment of the velocity by the lineola PO; and with the centre C, and rectangular afymptotes CA, CH, defcribe any Hyperbola BNS, meeting the erected perpendiculars AB, KN, LO in B, N, and O. Becaufe AK is as AP, the moment KL of the one will be as the moment 2 APO of the other, that is, as APXKC; for the increment PQ of the velocity is (by Law 2.) proportional to the generating force KC. Let the ratio of KL be compounded with the ratio of KN, and the rectangle KL KN will become as AP× KC KN; that is, (because the rectangle KC × KN is given) as AP. But the ultimate ratio of the hyperbolic area KNOL to the rectangle KLX KN becomes, when the points K and L coincide, the ratio

of

of equality. Therefore that hyperbolic evanefcent area is as AP. Therefore the whole hyperbolic area ABOL is compofed of particles KNOL which are always proportional to the velocity AP; and therefore is itself proportional to the fpace defcribed with that velocity. Let that area be now divided into equal parts, as ABMI, IMNK, KNOL, &c. and the abfolute forces AC, IC, KC, LC, &c. will be in a geometrical progreffion. O. E. D. And by a like reafoning, in the afcent of the body, taking, on the contrary fide of the point A, the equal areas AB mi, imnk, knol, &c. it will appear that the abfolute forces AC, ic, kC, IC, &c. are continually proportional. Therefore if all the spaces in the afcent and descent are taken equal; all the abfolute forces IC, kC, iC, AC, IC, KC, LC, &c. will be continually proportional.

O.E.D.

COR. 1. Hence if the space defcribed be expounded by the hyperbolic area AB NK; the force of gravity, the velocity of the body, and the refiftance of the medium, may be expounded by the lines AC, AP, and AK refpectively; and vice versa.

COR. 2. And the greateft velocity, which the body can ever acquire in an infinite defcent, will be expounded by the line AC.

COR. 3. Therefore if the refiftance of the medium anfwering to any given velocity be known, the greatest velocity will be found, by taking it to that given velocity in a ratio fubduplicate of the ratio which the force of gravity bears to that known refiftance of the medium.

PROPOSITION IX. THEOREM VII. Suppofing what is above demonftrated, I fay that if the tangents of the angles of the fector of a circle, and of an hyperbola, be taken proportional to the velocities, the radius being of a fit magnitude; all the time of the afcent to the highest place will be as the Jector of the circle, and all the time of defcending from the highest place as the fector of the hyperbola. Pl. 2. Fig. 2.

To the right line AC, which expresses the force of gravity, let AD be drawn perpendicular and equal. From the centre D with the femidiameter AD defcribe as well the quadrant At E of a Circle; as the rectangular Hyperbola AVZ, whofe axe is AX, principal vertex A, and afymptote DC. Let Dp, DP be drawn; and the circular fector At D will be as all the time of the afcent to the highest place; and the hyperbolic fector ATD as all the time of defcent from the higheft place: If fo be that the tangents Ap, AP of thofe fectors be as the velocities. Fig. 2.

CASE 1. Draw Dvg cutting off the moments or leaft particles Dv and q Dp, defcribed in the fame time, of the lector ADt and of the triangle ADp. Since thofe particles (because of the common angle D) are in a duplicate ratio of the fides, the particle t Dv will be qDpxt D2 PD2

as

that is, (because t D is given) as qDp

PD2

But p D is AD2 --Ap2, that is, ADAD × Ak, or ADxCk; and q Dp is ADxpq. There

foret Dv, the particle of the fector, is as

is, as the leaft decrement pq of the velocity directly, and the force Ck, which diminishes the velocity, inverlely; and therefore as the particle of time anfwering to the decrement of the velocity. And, by compofition, the fum of all the particles & Dv in the lector ADt, will be as the fum of the particles of time anfwering to each of the loft particles pq, of the decreafing velocity Ap, till that velocity, being diminished into nothing, vanishes; that is, the whole fector ADt is as the whole time of afcent to the highest place. Q.E. D.

CASE 2. Draw DOV cutting off the leaft particles TDV and PDQ of the lector DAV, and of the triangle DAQ; and thefe particles will be to each other as DT to DP2, that is, (if TX and AP are parallel) as DX to DA2 or TX2 to AP2; and, by divifion, as DX2-TX2 to DA2-AP2. But, from the nature of the hyperbola, DX-TX2 is AD2; and, by the fuppofition, AP is ADX AK. Therefore

the particles are to each other as AD2 to AD2 — ADX AK; that is, as AD to AD-AK or AC to CK: and therefore the particle TDV of the fector is PDQX AC

CK

PO

; and therefore (because AC and AD are

PQ; that is, as the increment of the velo

given) as CK

city directly, and as the force generating the increment inversely; and therefore as the particle of the time anfwering to the increment. And, by compofition, the fum of the particles of time, in which all the particles PO of the velocity AP are generated, will be as the fum of the particles of the fector ATD; that is, the whole time will be as the whole fector. Q. E. D.

COR. I. Hence if AB be equal to a fourth part of AC, the space which a body will defcribe by falling in any time will be to the space which the body could describe, by moving uniformly on in the fame time