the line first mentioned would fall within the circle; which is contrary to the supposition of motion in a curve, which we have represented by straight lines indeed in the first instance, but of evanescent magnitude, and leaving no definite distance between themselves and the curve.

XXIII. C.

And now my third proposition is that which I Prop. originally undertook to demonstrate. Supposing the bodies to describe respectively the arcs AB and CD (which, though here represented of a convenient magnitude, must be conceived as indefinitely small) in one indefinitely small portion of time, the chords or straight lines AB and CD, subtending those arcs, may, without any possibility of error, be regarded as coinciding with the arcs themselves. Now if AG and CH be drawn at right angles to SA and SC, they will represent the directions in which the bodies would move independently of any centripetal force; (according to Prop. XXIII. B;) and as the centripetal force acts in the directions AS, CS, each body will describe the diagonal of a rectangle; but the diagonals actually described are AB and CD; and by drawing BG, BE, respectively parallel to AS and AG, and DH, DF, respectively parallel to CS and CH, the rectangles will be completed; and then AE and CF will be the spaces through which the bodies would fall, in the supposed indefinitely small portion of time, by the sole operation of the centripetal forces, and therefore (by Prop. XXIII. A.)

will be to each other in the ratio of those forces. Again the rectangle AK, AE, is equal to the square of AB; and the rectangle CI, CF, is equal to the square of CD: that is, the square of the chord or arc AB, divided by AK, is equal to AE; and the square of the chord or arc CD, divided by CI, is equal to CF. Therefore the centripetal force at A is to the centripetal force at C, as the square of the indefinitely small arc AB, divided by AK, to the square of the indefinitely small contemporaneous arc CD, divided by CI. And the same proportion, it is evident, will still be preserved, if the radii or semidiameters of the circles are substituted for their entire diameters as divisors. And the same proportion will also be preserved, if arcs of a definite magnitude, described in the same definite portion of time, (as a second, an hour, &c.) be substituted for the indefinitely small arcs described in one indefinitely small portion of time. For the velocities being uniform, the arcs are increased proportionally with the time; and if the arcs are increased proportionally, so are their squares.

A. I do not quite perceive the certainty of your last assertion.

B. When any two variable quantities are in a certain ratio the one to the other, which ratio they retain through all their changes; if you multiply the one always by one certain number, and the other by one other certain number, the ratio

will indeed be altered, but that altered ratio will also continue the same through all changes of the original quantities.

A. That, to be sure, is evident.

B. But now suppose these multipliers themselves to vary, only retaining always the same ratio the one to the other, will not the same consequence follow? Can it make any difference, for instance, as to the mutual ratio of the products, whether you multiply the one quantity by three, and the other by seven, or the one by six, and the other by fourteen, or the one by nine, and the other by twenty-one?

A. No; I allow the case to be the same.

B. But this you must see is done, when the squares of quantities which always retain the same mutual ratio are taken instead of the quantities themselves. So that though the ratio existing between any two quantities is not the same with the ratio between their squares; yet, if those quantities be so augmented or diminished as always to preserve one and the same ratio between themselves, their squares will also be so augmented or diminished as always to preserve one and the same ratio between themselves. And now, if you are satisfied of the truth of this proposition, I may proceed to draw some important consequences from it; but it will be best to reserve them for our next meeting.

B. THE first thing to be done, in order to draw the desired consequences from our last proposition, is in some degree to alter its enunciation. Since then the arcs described with uniform motion in the same time represent the velocities of the bodies, we may say that the centripetal forces, in different circles having the same centre, are as the squares of the velocities, divided by the radii; unless indeed you are staggered by the incomprehensibleness of a squared velocity.

A. No; I see that the expression may be justified in the arithmetical sense.

B. And as the periodical times, or durations of entire revolutions in the several circles, would, if the velocities were equal in all, be proportionate to the circumferences of those circles; but are halved where the velocities are doubled, trisected where the velocities are tripled, and so forth; we may say that the periodical times are as the circumferences divided by the velocities; or (since the circumferences of circles are proportionate to their radii) as the radii divided by the velocities. Now you know that the quotient of one quantity divided by another is properly represented by a fraction; the numerator, or quantity above the

or

VV

R

R

line, being the dividend, and the denominator, or quantity below the line, being the divisor. The variable quantity therefore, to which the periodical time is proportionate, may be represented by the fraction; and the variable quantity to which we have found the centripetal force to be proportionate, by the fraction V times V divided by R Now it is evident from the nature of a fraction, that if you would multiply it by any quantity, you may either multiply the numerator, or divide the denominator by that quantity, as you find most convenient. And so, And so, if you would divide the fraction, you may effect it either by division of the numerator, or by multiplication of the denominator. And in the present instance it is obvious that by multiplying, the first of these fractions, three times by V, and dividing it twice by R, we shall convert it into the second, ; and therefore the force may be said to vary as VRR or (dividing both the dividend and divisor by V) that is, as R twice multiplied by. But to multiply by is the same thing as to divide by V ; for, as multiplying any quantity by the numerator, and dividing it by the denominator, is the same thing as multiplying it by their quotient, which is the fraction itself; so, in order to restore the original quantity, (which after such multiplication is the work of division,) it is evident that the operation must be reversed. The proportion therefore will not be altered, if, instead of

as

R

RVV

RR

V

R

D

R

VV

R

RVVV