in the corresponding fituations, the relation of their measures may be expressed by a proportional equation, according to Def. 2. 3. If three variable physical quantities are fo connected, that one of them is increafed or diminished, in proportion as both the others are increased or diminished; or, if the magnitudes of one of them in any two fituations, have a ratio, which is compounded of the ratios of the magnitudes of the other two, in the corresponding fituations; the relation of the measures of these three may be expreffed by a proportional equation, according to Def. 3. 4. In like manner may the relations of other combinations of phyfical quantities be expreffed, according to Def. 4. And when these proportional equations are obtained, by reasoning with regard to them, according to the preceding propofitions, new relations of the phyfical quantities may be deduced. 2. Examples of Phyfical Problems. The ufe of algebra, in natural philofophy, may be properly illuftrated by fome examples X examples of phyfical problems. The folution of such problems must be derived from known physical laws, which, though ultimately founded on experience, are here af fumed as principles, and reasoned upon mathematically. The experiments by which the principles are afcertained admit of various degrees of accuracy; and on the degree of phyfical accuracy in the principles will depend the phyfical accuracy of the conclufions mathematically deduced from them. If the principles are inaccurate, the conclufions muft, in like manner, be innaccurate; and, if the limits of inaccuracy in the principles can be ascertained, the correfponding limits, in the conclufions derived from them, may likewife be calculated. EXAMPLE I. Let a glafs tube, 30 inches (a) long, be filled with mercury, excepting 8 inches (b); and let it be inverted, as in the Torricellian experiment, fo that the 8 inches of common air may rife to the top: It is required to find at what height the mercury will remain fufpended, the mercury in the barometer barometer being at that time 28 inches (d) high. The folution of this problem depends upon the following principles: 1. The preffure of the atmosphere is measured by the column of mercury in the barometer; and the elastic force of the air, in its natural state, which refifts this preffure, is therefore measured by the fame column. 2. In different ftates, the elaftic force of the air is reciprocally as the spaces which it occupies. 3. In this experiment, the mercury which remains fufpended in the tube, together with the elastic force of the air in the top of it, being a counterbalance to the preffure of the atmosphere, may therefore be expreffed by the column of mercury in the barometer. Let the mercury in the tube be x inches, the air in the top of it occupies now the fpace ax; it occupied formerly 6 inches, b and its elastic force was d inches of mercury: Now, therefore, the force must be bd inches (2.). Therefore (a—x : b :: d : ),—x bd (3.)x+ =d. This reduced, and put AX ting a+d=2m, the equation is x2-2mx -bd-ad. This refolved gives x=m±√m2+bd—ad. In numbers x=44 or 14. One of the roots 44 is plainly excluded in this cafe, and the other 14 is the true anfwer. If the column of mercury x, fufpended in the tube, were a counterbalance to the preffure of the atmosphere, expreffed by the height of the barometer d, together with the measure of the elaftic force of b inches of common air in the fpace xa, that is, the experiment in this queftion does not admit of fuch a fuppofition. EXAMPLE II. The distance of the earth and moon (d,) and their quantities of matter (t, 1,) being given, to find the point of equal attraction between them. Let Let the distance of the point from the earth be x, its distance from the moon will be therefore d-x. But gravitation is as the matter directly, and as the square of the distance inversely; therefore the earth's attraction is as; and the moon's attrac tion is as. But these are here equal; therefore, In round numbers, let d=60 femidiameters of the earth, t=40, 1, then x=52 femidiameters nearly. There is another point beyond the moon at which the attractions are equal, and it would be found by putting the square root of d2 to be x-d, which, in this cafe, would be a positive |