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progression, there will be as many lines Xr in an arithmetical progression, And hence the curve DraF is easily delineated by the table of logarithms.
Cor. 3. If a parabola be constructed to the vertex D (PI. 1. Fig. 5), and the diameter DG produced downwards, and its latus rectum is to 2 DP as the whole resistance at the beginning of the motion to the gravitating force, the velocity with which the body ought to go from the place D, in the direction of the right line DP, so as in an uniform resisting medium to describe the curve DraF, will be the same as that with which it ought to go from the fame place D, in the direction of the same right line DP, so as to describe a parabola in a non-refifting medium. For tlie latus rectum of this parabola, at the
tGT very beginning of the motion, is and Vr is
N DR X Tt 2N
But a right line, which, if drawn, would touch the hyperbola GTS in G, is parallel to DK, and therefore Tt CK X DR
QB x DC is and N is
And therefore Vr is DC
CP DR2 X CK X CP equal to
that is (because DR and DC, 2DC2 X QB
DV2 X CK X CP DV and DP are proportionals), to
2DP2 x QB the latus rectum comes out
that is (because Vr
CK X CP
2DP X DA QB and CK, DA and AC are proportional),
x therefore is to ODP as DP X DA to CP X AC; that is, as the resistance to the gravity. Q.E.D.
Cor. 4. Hence if a body be projected from any place D (Pl. 1, Fig. 4) with a given velocity, in the direction of a right line DP given by pofition, and the resistance of the medium, at the beginning of the motion, be given, the curve DraF, which that body will defcribe, may be found. For the velocity being given, the latus rectum of the parabola is given, as is well known. And taking 2DP to that latus rectum, as the force of gravity to the resisting force, DP is also given.
Then cutting DC in A, fo that CP * AC may be to DP DA in the same ratio of the gravity to the resistance, the point A will be given. And hence the curve DraF is also given.
Cor. 5. And, on the contrary, if the curve DraF be given, there will be given both the velocity of the body and the resistance of the medium in each of the places r. For the ratio of CP X AC to DP X DA being given, there is given both the resistance of the medium at the beginning of the motion, and the latus rectum of the parabola ; and thence the velocity at the beginning of the motion is given also. Then from the length of the tangent rL there is given both the velocity proportional to it, and the resistance proportional to the velocity in any place r.
Cor. 6. But since the length 2DP is to the latus rectum of the parabola as the gravity to the resistance in D; and, from the velocity augmented, the resistance is augmented in the same ratio, but the latus rectum of the parabola is augmented in the duplicate of that ratio, it is plain that the length 2DP is augmented in that simple ratio only; and is therefore always proportional to the velocity ; nor will it be augmented or diminished by the change of the angle CDP, unless the velocity be also changed.
Cor. 7. Hence appears the method of determining the curve DraF (Pl. 1, Fig. 6) nearly, from the phænomena, and thence collecting the resistance and velocity with which the body is projected. Let two similar and equal bodies be projected with the same velocity, from the place D, in different angles CDP, CDp; and let the places F, f, where they fall upon the horizontal plane DC, be known. Then taking any length for DP or Dp, suppose the resistance in D to be to the gravity in any ratio whatsoever, and let that ratio be expounded by any length SM. Then, by computation, from that afsumed length DP, find the lengths DF, Df; and from the ratio Ff DF
found by calculation, fubduct the fame ratio as found by experiment; and let the difference be expounded by the perpendicular MN. Repeat the same a second and a third time, by assuming always a new ratio SM of the resistance to the gravity, and collecting a new difference MN. Draw the affirmative differences on one side of the right line SM, and the negative on the other side; and through the points N,N,N, draw a regular curve NNN, cutting the right line SMMM in X, and SX will be the true ratio of the resistance to the gravity, which was to be found. From this ratio the length DP is to be collected by calculation; and a length, which is to the assumed length DP as the length DF known by experiment to the length DF just now found, will be the true length DP. This being known, you will have both the curve line DraF which the body describes, and also the velocity and refistance of the body in each place.
SCHOLIUM. But, yet, that the resistance of bodies is in the ratio of the velocity, is more a mathematical hypothesis than a physical one. In mediums void of all tenacity, the resistances made to bodies are in the duplicate ratio of the velocities. For by the action of a fwifter body, a greater motion in proportion to a greater velocity is communicated to the fame quantity of the medium in a less time, and in an equal time, by reason of a greater quantity of the disturbed medium, a motion is communicated in the duplicate ratio greater; and the refiftance (by law 2 and 3) is as the motion communicated. Let us, therefore, see what motions arise from this law of re fiftance.
SECTION II. Of the motion of bodies that are repíted in the duplicate ratio
of their velocities. PROPOSITION V. THEOREM III. If a body is resisted in the duplicate ratio of its velocity, and
moves by its innate force only through a similar medium; and the times be taken in a geometrical progreson, proceeding from lefs to greater terms: I say, that the velocities at the beginning of each of the times are in the same geometrical progresion inversely; and that the spaces are equal, which are described in each of the times. (Pl. 1, Fig. 7.)
For fince the resistance of the medium is proportional to the square of the velocity, and the decrement of the velocity is proportional to the resistance: if the time be divided into in
numerable equal particles, the squares of the velocities at the beginning of each of the times will be proportional to the differences of the same velocities. Let those particles of time be AK, KL, LM, &c. taken in the right line CD; and erect the perpendiculars AB, Kk, LI, Mm, &c. meeting the hyperbola Bklm G, defcribed with the centre C, and the rectangular afymptotes CD, CH, in B, k, l, m, &c.; then AB will be to Kk as CK to CA, and, by division, AB - Kk to Kk as AK to CA, and, alternately, AB — Kk to AK as Kk to CA; and therefore as AB x Kk to AB x CA. Therefore fince AK and AB x CA are given, AB — Kk will be as AB x Kk; and, lastly, when AB and Kk coincide, as AB?. And, by the like reafoning, Kk- LI, LI -- Min, &c. will be as Kk?, Ll, &c. Therefore the squares of the lines AB, Kk, LI, Mm, &c. are as their differences; and, therefore, fince the squares of the velocities were shewn above to be as their differences, the progreflion of both will be alike. This being demonftrated it follows also that the areas described by these lines are in a like progression with the spaces described by these velocities. Therefore if the velocity at the beginning of the first time AK be expounded by the line AB, and the velocity at the beginning of the second time KL by the line Kk, and the length described in the first time by the area AKIB, all the following velocities will be expounded by the following lines LI, Mm, &c. and the lengths described, by the areas KI, Lm, &c. And, by composition, if the whole time be expounded by AM, the sum of its parts, the whole length described will be expounded by AMmB the sum of its parts. Now conceive the time AM to be divided into the parts AK, KL, LM, &c. fo that CA, CK, CL, CM, &c. may be in a geometrical progression; and those parts will be in the same progression, and the velocities AB, Kk, LI, Mm, &c. will be in the same progression inversely, and the spaces defcribed Ak, KI, Lm, &c. will be equal. Q.E.D.
Cor. 1. Hence it appears, that if the time be expounded by any part AD of the asymptote, and the velocity in the beginning of the time by the ordinate AB, the velocity at the end of the time will be expounded by the ordinate DG; and the whole space described by the adjacent hyperbolic
area ABGD; and the space which any body can describe in the same time AD, with the first velocity AB, in a non-refifting medium, by the rectangle AB X AD.
Cor. 2. Hence the space described in a refifting medium is given, by taking it to the space described with the uniform velocity AB in a non-resisting medium, as the hyperbolic area ABGD to the rectangle AB X AD.
CoR. 3. The resistance of the medium is also given, by making it equal, in the very beginning of the motion, to an uniform centripetal force, which could generate, in a body falling through a non-resisting medium, the velocity AB in the time AC. For if BT be drawn touching the hyperbola in B, and meeting the asymptote in T, the right line AT will be equal to AC, and will express the time in which the first resistance, uniformly continued, may take away the whole velocity AB.
Cor. 4. And thence is also given the proportion of this refiftance to the force of gravity, or any other given centripetal force.
Cor. 5. And, vice versa, if there is given the proportion of the resistance to any given centripetal force, the time AC is also given, in which a eentripetal force equal to the resistance may generate any velocity as AB; and thence is given the point B, through which the hyperbola, haviņg CH, CD for its asymptotes, is to be described; as also the space ABGD, which a body, by beginning its motion with that velocity AB, can describe in any time AD, in a similar resisting medium.
PROPOSITION VI. THEOREM IV. Homogeneous and equal spherical bodies, opposed by resistances
that are in the duplicate ratio of the velocities, and moving on by their innate force only, will, in times which are reciprocally as the velocities at the beginning, describe equal spaces, and lose parts of their velocities proportional to the wholes. (Pl. 1, Fig. 8.)
To the rectangular asymptotes CD, CH describe any hyperbola BbEe, cutting the perpendiculars AB, ab, DE, de in B, b, E, e; let the initial velocities be expounded by the perpendiculars AB, DE, and the times by the lines Aa, Dd.