back. By contracting or expanding the fins, they alfo affift him in afcending and defcending: by inclining his tail obliquely, and turning it a little from an erect pofition to one fide, it helps him to rife and fall. Fish fwim but flow, and foon tire; yet some are said to swim feventy or eighty yards in a minute.

Brutes fwim naturally; for they are specifically lighter than water, and require but a fmall part of their head out for breathing. It is alfo easy to them; for they use their legs in fwimming after the fame manner as they do in walking.

Men do not fwim naturally, though they are fpecifically lighter than water; for their heads are very large, and require to be almoft out of the water for breathing, and their way of striking has no analogy to walking. Men attain the art of fwimming by practice and industry. The art confifts in ftriking the water alternately with the feet and hands; which, like oars, row him forward. When he ftrikes with his hands, he neither keeps the palm parallel, nor perpendicular to the horizon, but inclined; and his hands ftriking the water obliquely, the refiftance of the water moves him partly upward and partly forward. Whilft his hands are ftriking, he gradually draws up his feet; and when the ftroke of his hands is over, he ftrikes with his feet, by extending his legs, and pufhing the foles of his feet againft the water; and while he ftrikes with his legs, he brings about his arms for a new ftroke, and fo on alternately; keeping the body fomewhat oblique, that he may more easily erect his head, and keep his mouth above water.

LECTURE XXVII.

OF ACCELERATED MOTION.

THO

HOUGH the most unlettered ruftic is fenfible that the fall of a stone is to be dreaded in proportion to the height from whence it defcends; that if it falls from a place a foot above his head, it is not fo likely to be fatal as if it fell from the house-top; yet the law of acceleration, in falling bodies, was not difcovered till the time of GaFileo: to inveftigate the effect of gravity on fuch Bodies was referved for him, who was one of the greatest ornaments of the age in which he lived.

Motion is faid to be accelerated, if it's velocity continually increafes; to be uniformly accelerated, if it's velocity increafes equally in equal times.

Motion is faid to be retarded, if it's velocity continually decreases; and to be uniformly retarded, if it's velocity decreases equally in equal times.

In our general notions of acceleration or retardation, we abftract our attention from all varieties, and they are fuppofed uniform: that is, we fuppofe that the velocities increase or diminish at the fame rate with the times, or that the changes of velocity are proportioned to the times in which they are acquired.

Accelerations and retardations may be confidered as quantities, and are measured by the changes of velocity, which are uniformly acquired in the fame or equal times; they are therefore proportional to the changes of velocity directly, and

to

to the time in which they are uniformly acquired, inverfely.

If the velocities do not change at the fame rate. with the time, the acceleration or retardation is not conftant.

If you fuppofe a body to be put in motion by a fingle impulfe, and moving uniformly, to receive a new impulse in the fame direction, it's velocity will be augmented, and it will go on with the augmented velocity. If at each inftant of it's motion it receives a new impulfe, the velocity will be continually increafing; and if this impulfe is always, equal, and acts in equal times, the velocity will be uniformly accelerated.

The force, which at every inftant gives this new impulse to a body, is called the accelerating force.

For the fame reafon, if a body had at first a certain velocity, and lofes equal parts at each equal inftant, by new impulfions acting in a direction exactly contrary to it's motion, it is faid to be uniformly retarded.

Every body left to itself defcends with an accelerated motion, and a direction which tends to the center of the earth. The force caufing this acceleration, and occafioning this tendency, is called gravity. It is a force always prefent; and which, conftantly acting on bodies, uniformly accelerates their motion.

All that we can obferve in motion, is the space deferibed, and the time in which it is defcribing. From obfervations on thefe, we infer the proportions of the changing forces. Uniformly accelerated or retarded motion is therefore the indication of an invariable or conftant force. The changes, therefore, of velocity in any time, by a force varying according to any law, is the proper measure

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measure of the accumulated or whole action of the force during this time.

I fhall now endeavour to explain the law of bodies falling with an accelerated motion by the force of gravity. The accelerating force or gravity decreases, as the fquares of the diftances increase; but as the greateft diftance from which we can obferve the force of falling bodies, bears no proportion to that from the center of the earth, we may confider gravity as acting without any decrease, or, rather, that it's force is the fame from the beginning to the end of the fall.

The nature of all accelerating forces is to act continually on the moving body, and to be fucceffively impelling it by an infinity of fmall ftrokes. But the accelerating force being thus repeated as often as there are inftants in the given time, is proportional to the product of the weight by the last acquired velocity.

As gravity acts uniformly on all bodies at an equal distance from the center of the earth, let us fuppofe the time of the defcent to be divided into a number of equal parts, infinitely fmall; the impreffion of gravity will occafion the body to defcend towards the earth in the firft fmall inftant. Now if you fuppofe gravity to cease to act, the body will defcend uniformly, in confequence of it's first impreffion, with an infinitely fmall velocity; but, on the contrary, if at the fecond inftant, a new and equal impulfe is added to the former; it's velocity, in the second inftant, will double that of the firft; in the third, by the continued impulfe, three times; four times in the fourth; and fo on: for the impreffion made in the preceding inftants are not at all impaired by thofe that follow, but are, if I may be allowed the expreffion, heaped one upon another; confequently, as the body is fuppofed to

receive

receive a new impreffion every moment of it's fall the velocity increases, as the moments, or time increases.

In the right-angled triangle CAB, fig. 12, pl. I, which opens wider, by equal degrees, from the point A towards BC, the base of the triangle expreffes a motion uniformly accelerated.

The leg A B expreffes the time in which this acceleration happens, and this time is divided into four equal parts, or moments, as A 1, 12, 23, 3 E. The fmall lines in the triangle A 1 k, repeated at equal intervals, and increafing in length by equal degrees, denote equal accelerations of the velocity from the inftant in which a body begins to fall, and the fmall equal lines in the fquare 1 m, denote an equal velocity during the time expreffed by the fide 1, 2.

These things being kept in mind, if you attend to the diagram, it will open to you all the confequences of thefe pofitions.

The linek will reprefent the velocity acquired by a falling body in the first moment of time; 21, the velocity acquired at the end of the fecond moment of time; and 30, the velocity at the end of the third; and fo on.

If the body, during the fecond moment of time, should retain the velocity 1k, which it had acquired at the end of the first, it will defcribe the fquare furface 1, 2 m k; for this furface is generated by a continual repetition, or motion, of the line 1k, during 12; as the area of the triangle a 1 k is defcribed by an uniformly increafing velocity during the time A 1. But the area of the fquare is manifeftly double the area of the triangle; whence it appears, that a body moving on, during a fecond moment, with the velocity acquired at the end of the firft, will fall twice as far in the fe

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cond