Corol. 1. Hence the ofcillations in the cycloid are all performed in equal times; for they are all in the fame ratio to the time in which a body falls thro' the diameter D V. If therefore a pendulum ofcillates in a cycloid, the time of the ofcillation in any arc is equal to the time of the ofcillation in the greatest arc B VA, and the time in the least arc is equal to the time in the greatest.

Corol. 2. The cycloid may be confidered as coinciding, in v, with any fmall arc of a circle defcribed from the centre c; paffing thro' v; and the time in a small arc of fuch a circle will be equal to the time in the cycloid; and hence is understood why the times in very little arcs are equal, because these little arcs may be confidered as portions of the cycloid as well as of the circle.

Corol. 3. The time of a complete ofcillation in any little arc of a circle, is to the time in which a body would fall thro' half the radius; as the circumference of a circle, to its diameter: and fince the latter time is half the time in which a body would fall thro' the whole diameter, or any chord, it follows that the time of an ofcillation in any little arc, is to the time in which a body would fall thro' its chord; as the femicircle, to the diameter.

Suppose N v a small arc of the circle defcribed from the centre c; then the time in the arc N V is fo far from being equal to the time in the chord N v, even when they are supposed to be evanefcent, that the laft ratio of these times is that of the circumference of a circle to four times the diameter: and hence an error in several mechanical writers is to be corrected, who, from the equality of the evanefcent arcs and Q3

their

Book II. their chords, too rafhly conclude the time of a fall of a body in any of thefe arcs equal to the time of the fall of a body in their chords.

Corol. 4. The times of the ofcillations in cycloids, or in fmall arcs of circles, are in a fubduplicate ratio of the lengths of the pendulums. For the time of the oscillation in the arc LVP is in a given ratio to the time of the fall thro' DV, which time is in the fubduplicate ratio of the fpace D v, or of its double cv the length of the pendulum.

Corol. 5. But if the bodies that ofcillate be acted on by unequal accelerating forces, then the ofcillations will be performed in times that are to one another in the ratio compounded of the direct fubduplicate ratio of the lengths of the pendulums, and inverfe fubduplicate ratio of the accelerating forces: because the time of the fall thro' D v is in the fubduplicate ratio of the space D v directly and of the force of gravity inverfely; and the time of the ofcillations is in a given ratio to that time. Hence it appears, that if ofcillations of unequal pendulums are performed in the fame time, the accelerating gravities of these pendulums must be as their lengths; and thus we conclude that the force of gravity decreases as you go towards the equator; fince we find that the lengths of pendulums that vibrate feconds are always lefs at a lefs diftance from the equator.

Corol. 6. From this propofition we learn how to know exactly what fpace a falling body defcribes in any given time for finding, by experiment, what pendulum ofcillates in that time, the half of the length of the pendulum will be to the space required, in the duplicate ratio of the diameter to the circumference; becaufe fpaces defcribed by a falling body,

from

from the beginning of its motion, are as the fquares of the times in which they are defcribed; and the ratio of the times, in which these spaces are described, is that of the diameter to the circumference: and thus Mr. Huygens demonftrates that falling bodies, by their gravity only, defcribe 15 Parifian feet and I inch in a fecond of time.

Schol. That it may be understood how the time in a small arc is not the fame with that in its chord, tho' the evanefcent arc is equal to its chord, we may here demonftrate, that if vk and Nk be two planes touching the arc NV in v and N. Tho' the evanefcent chord Nv be equal to the fum of thefe tangents v k and Nk, yet the time in the chord is to the time in these tangents as 4 to 3.

By Cor. 1. Lem. 3. the time in n k is to the time in NV as Nk to N v, or as 1 to 2; but k v being horizontal, the motion in k v must be uniform, and it will be described by that uniform motion in half the time the body falls from N to k: therefore if the time in which kv is defcribed uniformly be called T, the time in which Nk is defcribed will be 2 T, and the time in which the chord Nv will be defcribed will be 4T and confequently the time in which a body would fall along the two tangents, is to the time in which it would defcribe the chord, as 3 to 4.

BOOK III.

Gravity demonftrated by analyfis.

CHA P. I.

Of the theory of gravity as far as it appears to bave been known before Sir Ifaac Newton.

"F

AROM experiments and obfervation alone, we are enabled to collect the hiftory of nature, or describe her phænomena. By the principles of geometry and mechanics, we are enabled to carry on the analyfis from the phænomena to the powers or causes that produce them; and, by proceeding with caution, we may be fatisfied that our foundations are well laid, and that the fuperftructure raised upon them is secure. The firtt views which philofophers had of nature were no better than those of the vulgar, being the immediate fuggeftions of fenfe. But by comparing thefe together, examining the nature of the fenfes themfelves, correcting and affifting them; and by a juft application of geometrical and mechanical principles, the scheme of nature foon appears very different to a philofopher from that which is presented to a vulgar eye. At first fight, the furface of the earth appears of an unbounded extent, and of a moft irregular form; while all the reft of the univerfe, the clouds, meteors, moon, fun, and stars of all forts, appear in one concave furface bent towards the earth. This was the opinion concerning the fyftem that most

com

commonly prevailed at first, while their imagination, influenced by fuch prejudices, made men fancy that they faw and heard things impoffible. Thus the Roman poet represents their army when in Portugal (the western boundary of the great continent) as hearing the fun enter with a hiffing noife into the

ocean,

Audiit berculeo ftridentem gurgite folem.

LUCAN.

while other travellers have talked of a vast cavity in the most remote parts of the eaft, from whence the fun was heard to iffue every morning with an unfufferable noise. But philofophers foon difcovered that the earth was not of an unbounded extent, but of a globular form; and that the meteors, planets, and itars, were not confined to one concave furface, but difperfed in Space at very different distances; that their real magnitudes and motions are very different from their apparent ones, and are not to be deduced from the appearances in any one place, but from views taken from divers points of fight, compared together by geometrical principles.

2. As our analysis of the fyftem must be founded upon the real figures, magnitudes and motions, of the bodies of which it is compofed; so we shall have an excellent inftance of the method of proceeding by analysis and fynthefis if we defcribe in what manner we are enabled, from the apparent phænomena, to deduce an account of the real, without the knowledge of which our enquiries into the powers or causes that operate in nature must be doubtful or erroneous. The knowledge of the difpofition and motions of the celeftial bodies muft precede a juft enquiry into their caufes. The former is more fimple, the latter more arduous; and the former will

pre