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how to define the spaces of the descent of a body, let fall from any given height, down to the centre, supposing the gravitation to increase, as in the fifth property ; but considering the smallness of height, to which any projectile can be made to ascend, and over how small an arch of the globe it can be thrown by any of our engines, we may well enough suppose the gravity to be equal throughout: and the descents of projectiles to be in parallel lines, which in reality are towards the centre, the difference being so small, as by no means to be discovered in practice. Propositions concerning the Descent of heavy Bodies, and the
Motion of Projects. -- [1685-6.] Prop. I. The velocities of falling bodies are proportional to the times, from the beginning of their falls. For the action of gravity being continual, in every space of time the falling body receives a new impulse, equal to what it had before, in the same space of time, received from the same power; for instance, in the first second of time, the falling body has acquired a velocity, which in that time would carry it to a certain distance, suppose 32 feet, and if there were no new force, it would descend at that rate with an equable motion; but in the next second of time,
gravity continually acting on it superadds a new velocity equal to the former ; so that at the end of two seconds, the velocity is double to what it was at the end of the first: and after the same manner may it be proved to be triple at the end of the third second, and so on. Therefore the velocities of falling bodies are proportional to the times of their falls. Q. E. D.
Prop. Il. The spaces described by the fall of a body are as the squares of the times, from the beginning of the fall. Let A B represent the time of the fall of a body; BC, perpendicular to AB, the velocity acquired at the end of the fall; and draw the line AC; then divide the line AB, representing the time, into as b many equal parts as you please, as b, b, b, b, &c. and through these points draw the lines bc, bcbc, bc, &c. parallel to BC. It is manifest that the several lines, bc, represent the several velocities of the falling B boily, in such parts of the time as Ab is of A B, by the former proposition. It is likewise evident, that the area A B C is the sum of all the lines bc; so that the area ABC represents thesum of all the velocities, between none and BC, supposed in
finitely many; which sum is the space descended in the time represented by A B. And, by the same reason, the areas Abc will represent the spaces descended in the times Ab; so then the spaces descended in the times AB, Ab, are as the areas of the triangles A BC, Abc, which by the 20th of the sixth of Euclid, are as the squares of their homologous sides A, B, Ab that is, of the times. Therefore the descents of falling bodies are as the squares of the times of their fall. Q. E. D.
Prop. III. The velocity, which a falling body acquires in any space of time is double to that with which it would have moved the space descended by an equable motion, in the same time. For, draw the line EC parallel to AB, and AE parallel to BC, in the same fig. 1., and complete the parallelogram, ABCE: it is evident that its area may represent the space a body moved equably with the velocity BC would describe in the time AB; and the triangle ABC represents the space described by the fall of a body, in the same time AB, by the second proposition. Now the triangle ABC is half the parallelogram ABCE, and consequently the space described by the fall is half what would have been described by an equable motion with the velocity BC, in the same time; therefore the velocity BC, at the end of the fall, is double to that velocity, which in the time A B would have described the space fallen, represented by the triangle ABC, with an equable motion. Q. E. D.
Prop. IV. All bodies on or near the surface of the earth, in their fall, descend in such a manner, as at the end of the first second of time, to have described 16 feet and one inch, London measure, and acquired the velocity of 32 feet two inches in a second. The length of the pendulum, vibrating seconds, being found 394 inches, the descent in a second will be found by the aforesaid analogy to be 16 feet one inch : and by the third prop. the velocity will be double thereto; and thus nearly it has been found by several experiments, which, by reason of the swiftness of the fall, cannot so exactly determine its quantity.
From these four propositions, all questions concerning the perpendicular fall of bodies are easily solved; and either the time, height, or velocity being assigned, the other two may be readily found. From them, likewise, is the doctrine of projectiles deducible, assuming the two following axioms; viz. That a body, put in motion, will move on continually in a right line with an equable motion, unless some other force or impediment intervene, by which it is accelerated, or retarded, or deflected. 2dly. That a body being agitated by two motions at a time, does by their compounded forces pass through the same points as it would do, if the two motions were divided and acted successively.
Account of the Trade Winds and Monsoons, observable in the
Seas between and near the Tropics. By E. HALLEY. The whole ocean may most properly be divided into three parts; viz. 1. The Atlantic and Ethiopic Sea. 2. The Indian Ocean. 3. The great South Sea, or the Pacific Ocean. And though these seas do all communicate by the south, yet, as to our present purpose of the trade winds, they are sufficiently separated by the interposition of great tracts of land.
I. In the Atlantic and Ethiopic Seas, between the tropics, there is a general easterly wind, all the year long, without any considerable variation, excepting that it is subject to be deflected some few points of the compass towards the north or south, according to the position of the place.
II. In the Indian Ocean the winds are partly general, as in the Ethiopic Ocean, and partly periodical, that is, half the year they blow one way, and the other half nearly on the opposite points, and these points and times of shifting are different in different parts of this ocean: the limits of each tract of sea subject to the same change or monsoon are certainly very hard to determine ; yet the following particulars may be relied on :
1. That between the latitudes of 109 and 30° south, as between Madagascar and New Holland, the general trade wind about the south-east by east, is found to blow all the year long, after the same manner as in the same latitudes of the Ethiopic Ocean.
2. That the aforesaid south-east winds extend to within 20 of the equator, during the months of June, July, August, &c. to November, at which time, between the south latitudes of 30 and 10°, being near the meridian of the north end of Madagascar, and between 20 and 12° south latitude, being near Sumatra and Java, the contrary winds from the northwest, or between the north and west, set in and blow for half the
year, viz. from the beginning of December till May; and this monsoon is observed as far as the Molucca isles.
3. That to the northward of 30 south latitude, over the whole Arabian or Indian Sea, and Gulf of Bengal from Sumatra to the coast of Africa, there is another monsoon, blowing from October to April, on the north-east points ; but in the
other half year, from April to October, on the opposite points of S. W. and W. S. W., and that with rather more force than the other, accompanied with dark rainy weather; whereas the north-east blows clear. It is likewise to be noted that the winds are not so constant, either in strength or direction, in the Gulf of Bengal, as they are in the Indian Sea, where a certain steady gale scarcely ever fails. It is also remarkable, that the south-west winds in these seas are generally more southerly on the African side, and more westerly on the Indian.
4. There is a tract of sea to the southward of the equator subject to the same changes of the winds, viz. near the African coast, between it and the island of Madagascar, and from thence northward, as far as the line ; wherein from April to October there is found a constant fresh S.S.W. wind, which, as you go more northerly, becomes still more and more westerly, so as to fall in with the W.S.W. winds mentioned before, in those months of the year to be certain to the northward of the equator.
5. That to the eastward of Sumatra and Malacca, to the northward of the line, and along the coast of Camboia and China, the monsoons blow north and south, that is, the northeast winds are much northerly, and the south-west much southerly. This constitution reaches to the eastward of the Philippine isles, and as far north as Japan. The northern monsoon setting-in in these seas in October or November, and the southern in May, blowing all the summer months. Here it is to be noted, that the points of the compass
from whence the wind comes in these parts of the world are not so fixed as in those lately described, for the southerly will frequently pass a point or two to the eastward of the south, and the northerly as much to the westward of the north i which seems occasioned by the great quantity of land interspersed in these seas.
6. That in the same meridians, but to the southward of the equator, being that tract lying between Sumatra and Java to the west, and New Guinea to the east, the same northerly and southerly monsoons are observed; but with this difference, that the inclination of the northerly is always towards the northwest, and of the southerly towards the south-east. But the points from which the winds blow are not more constant here than in the former, viz. variable five or six points. Besides, the times of the change of these winds are not the same as in the Chinese seas, but about a month or six weeks later.
7. That these contrary winds do not shift all at once ; but in some places the time of the change is attended with calms, in others with variable winds ; and it is particularly remarkable, that the end of the westerly monsoon on the coast of Coromandel, and the last two months of the southerly monsoon in the seas of China, are very subject to be tempestuous; the violence of these storms is such, that they seem to be of the nature of the West India hurricanes, and render the navigation of these parts very unsafe about that time of the year. These tempests are by our seamen usually termed, the breaking up of the monsoons. By reason of the shifting of these winds, such as sail in these seas are obliged to observe the seasons proper for their voyages; of which, if they miss, and the contrary monsoon sets in, they are forced to give up the hopes of accomplishing their intended voyage till the winds become favourable.
III. The third ocean, called Mare Pacificum, whose extent is equal to that of the other two, is that which is least known to our own or the neighbour nations; what navigation there is on it, is by the Spaniards, who go yearly from the coast of New Spain to the Manillas, and that only by one beaten track. What the Spanish authors say of the winds they find in their courses, and which is confirmed by the old accounts of Drake and Cavendish, and since by Schooten, who sailed the whole breadth of this sea in the south latitude of 15° or 16', is, that there is a great conformity between the winds of this sea and those of the Atlantic and Ethiopic seas ; that is, that to the northward of the equator, the predominant wind is between the east and north-east; and to the southward thereof, there is a constant steady gale between the east and south-east; and that on both sides the line, with so much constancy, that they scarcely ever need to attend the sails ; and with such strength, that it is usual to cross this vast ocean in ten weeks' time, which is about 130 miles a day; besides, it is said that storms and tempests are never known in these parts ; so that some have thought it might be as short a voyage to Japan and China, to go by the Straits of Magellan, as by the Cape of Good Hope.
Mathematical Principles of Natural Philosophy. By ISAAC
Newton, Lucasian Professor of Mathematics at Cambridge. Analysed by Dr. HALLEY.
This treatise is divided into three books, whereof the first two are entitled De Motu Corporum, the third De Systematę Mundi. The first begins with definitions of the terms made