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great study, to follow the discoveries which their disciples had made, by proceeding in the line which they themselves had pointed out. In this work, though a great number of ingenious men have been concerned, yet more is due to EULER than to any other individual. With indefatigable industry, and the resources of a most inventive mind, he devoted a long life entirely to the pursuits of science. Besides producing many works on all the different branches of the higher mathematics, he continued, for more than fifty years during his life, and for no less than twenty after his death to enrich the memoirs of Berlin, or of Petersburgh, with papers that bear, in every page, the marks of originality and invention. Such, indeed, has been the industry of this incomparable man, that his works, were they collected into one, notwithstanding that they are full of novelty, and are written in the most concise language by which human thought can be expressed, might vie in magnitude with the most trite and verbose compilations. The additions we have enumerated were made to the
mathematics; that which we are going to mention belongs to the mixt. It is the mechanical principle, discovered by D'Alembert, which reduces every question concerning the motion of bodies, to a case of equilibrium. It consists in this: If the motions, which the particles of a moving body, or a system of moving bodies, have at any instant, be resolved each into two, one of which is the motion which the particle had in the preceding instant, then the sum of all these third motions must be such, that they are in equilibrium with one another. Though this principle is, in fact, nothing else thar. the equality of action and reaction, properly explained, and traced into the secret process which takes place on the communication of motion, it has operated on science like one entirely new, and deserves to be considered as an important discovery. The consequence of it has been, that as the theory of equilibrium is perfectly understood, all problems whatever, concernining the motion of bodies, can be so far subjected to mathematical computation, that they can be expressed in fuxionary or differential equations, and the solution of them reduced to the integration of those equations. The full value of the proposition, however, was not understood, till La Grange published his Méchanique Analytique; the principle is there reduced to still greater simplicity; and the connexion between the pure and the mixt mathematics, in this quarter, may be considered as complete.
Furnished with a part, or with the whole of these resoursces, according to the period at which they arose, the mathematicians who followed Newton in the career of physical astronomy, were enabled to add much to his discoveries, and at last to complete the work which he so happily began. Out of the number who embarked in this undertaking, and to whorn science has many great obligations, five may be regarded as the leaders, and as distinguished above the rest, by the greatness of their achievements. These are CLAIRAUT, Euler, D'ALEMBERT, La GRANGE, and La Place. By their efforts it was
found, that, at the close of the last century, there did not remain a single phenomenon in the celestial motions, that was not explained on the principle of Gravitation as was taught by Newton, that forces propagated through an elastic medium diminish as the square of the distance, whether the cause be motion, as has been recently maintained.
The work which unites the application of all these discoveries is the Traité de Méchanique Céleste of La Place. The reasoning employed is every where algebraical ; and the various parts of the higher mathematics, the integral calculus, the method of partial differences and of variations, are from the first outset introduced, whenever they can enable the author to abbreviate or to generalize his investigations. No diagrams or geometrical figures are employed; and the reader must converse with the objects presented to him by the language of arbitrary symbols alone. The perfection of Algebra tends to the banishment of diagrams, and of all reference to them. La Grange, in his treatise of Analytical Mechanics, has no reference to figures, notwithstanding the great number of mechanical problems which he resolves. The resolution of all the forces that act on any point, into three forces, in the direction of three axes at right angles to one another, enables one to express their relations very distinctly, without representing them by a figure, or expressing them by any other than algebraic symbols. Thus Algebra, which was first introduced for the mere purpose of assisting geometry, and supplying its defects, has ended, as many auxiliaries have done, with discarding that science (or at least its peculiar methods) almost entirely. In those abstruse branches of mathematics, it must be confessed, that the French mathematicians have for years taken the lead of the English, but it may be hoped that the recent translation of Lacroix's work on the differential calculus by Messrs. Peacock and Herschel, and the writings of Messrs. Woodhouse and Ivory, will soon render these subjects familiar in England, and that, as in many former instances, we shall greatly improve on the invention of our neighbours.
ELEMENTS OF MATHEMATICS.
DEFINITIONS. 1. ALGEBRA is the science of Analysis, which, reasoning apon quantity or number by symbols, examines, in general, all the different methods and cases that can exist in the doctrine and calculation of numbers,
2. The symbols or characters adopted are the letters of the alphabet, which are called algebraic quantities, as a, b,
Notes.-1. These letters stand for numbers, and can, therefore, be applier to any thing to which numbers can be applied.
2. They differ from figures, because euch figure expresses a determinale nomber; but each of the letters stands for any number whatever, and, consequently, any thing which can be proved respecting any of the letters of symbols, is applicable to any number whatever.
3. Numbers are connected with the algebraic symbols in two different ways; as, 7x, or *®; signifying 7 times x, or the second power of x.
4. When the figure is put before the algebraic quantity, it is called the co-efficient, and shows how often the quantity is taken; as Sa, or 7x.
5. When ihe figure is put at the right hand corner of the algebraic quantity, it is called an index, or exponent, and depotes its power; as x', am, 2m, x", &c. where x2 shows the square, or second power of x; a' its cube, or third power ; on the mth power of x ; xk the nth power, &c.
Note. The mth or nth power of any quantity is a general expression for any power whatever.
6. If the figure be a fraction, as at, xt, mt, x*, it re. presents the root; thus, ce denotes the square root of *; at its cube root; at its mth root, at its nth root, &c.
Note. The mth or nth root of any quantity is a general expression for any moot wbatever.
7. Like quantities are composed of the same letters with the same indices, as a, 6a, 7ab, or 7b"x", 9b'x.
8. Unlike quantities consist of different letters, or of the same letters with different indices; as a and b, or 2a and a', or 3a'b', 4ab'c. /
9. Given quantities have their values all known, and are generally expressed by the early letters of the alphabet; as, u, b, c, d, &c.
10. Unknown quantities are those whose values are unknown, and are generally expressed by the final letters of the alphabet; as, X, Yor 2.
11. The sign + is read plus, or more. 12. The sign - is read minus, or less.
13. Simple quantities are composed of one term only; as, a, b, 6ab, 7 ar*, &c.
14. Compound quantities consist of several terms, connected by the sign plus or minus ; as, a+b, 7ax—3b, 3ab + b^c, &c.
15. Positive or affirmative qurintities are such as have the sign + before them; as, + a, + Gry.
16. Negative quantities are those which have the sign before them; as,
a, Oxy. 17. Like signs are all affirmative (+) or all negative ( - ).
18. Unlike signs are composed of affirmative (+), and negative (-) sigus.
19. A binomial quantity consists of two terms, as, a + b; a trinomial of three terms, as, a + b — c; and a quadrmomial of four, as, a + b − c + d, &c.
20. A residual quantity is a binomial, in which one of the terms is negative; as, a —
b. 21. A surd or irrational quantity has no exact root, as va. or 'a', or abt.
22. A rational quantity has no radical sign (w) or index annexed to it; as, a or ab. :23. The reciprocal of any quantity is that quantity inverted, or unity divided by it; as, is á, and of i
a' Further Explanation of Algebraic Characters. The sign + is employed to connect, or add, one quantity with another : it is, therefore, the symbol, or sign, of addition. The sign
indicates that the number before which it is placed, is to be taken from the number which precedes the symbol.' It is, therefore, the sign of subtraction. x or a dot (.) is the sign of multiplication.
of division. ::::.
of the square root.* ♡
of the cube root.
of equality, and read equal to. Chus, a + b shews that the number represented by b is to be added to that represented by a.
a-shews that the number represented by b is to be subtracted from that represented by a.
Roots are also expressed by fractional indices, as explained in Def. 6.
an b represents the difference of a and l when it is not known which is the greater.
a xb, or a,b, or ab, denotes the product of the numbers represented by a and b,
a+ b, or indicates that the number represented by a is to be divided by that denoted by b.
a:1:: c:d expresses, that a is in the same ratio, or proportion to l, that c is to d.
:=-6+ c exhibits an equation, shewing that x is equal to the difference of a and b, added to c.
(a+b)c, or a +bxc, is the product of the compound quantity a to multiplied by the simple quantity c; a+b + a-b, or (a+b) +
ato 12-1), or is the quotient of a +b divided by a-b; and the bar thus placed over two or more quantities, to connect them, is called a vinculum.
abor (a +1-c)' is the cube, or third power, of the quantity at-c.
5a indicates that the quantity a is to be caken 5 times ; likewise, 7 (li+c) is 7 times (b+c).
Note. The axioms of Geometry apply also to several branches of Algebra, and should be studied in connection with these definitions.
ADDITION. (24.) From the twofold division of algebraic quantities into positive and negative, like and unlike, there arise three cases of Addition, which must be separately considered.
Case I. To add like quantities with like signs. Rule. Add all the co-efficients, annex the common letter, or letters, and prefix the common sign.
Nole.--When a jeading quantity has no sign prefixed to it, the sign + plus is always understood ; and a quantity without any co-efficient is supposed to have 1, or unity before it. Thus a = once u (1a).
Examples.* 1 2
4 6a 3 at 2 %
7 22 + 2xy — 5 bc 53% + 5 xy— 1 8 a 2 at 3r
9 ro + 3xy 6 bc 3 2 + 2 xy— 3 9a 4 a + 4 x
8 bc 4 x + 6 xy-18 6 a + 91 — 0
х” + бу
bc x2 +
bc 6 x2 + 8 xy-14
12 a 2 a
Obs.Quantities with any kinds of exponents are, in all respects, to De considered as if they were represented by a single letter. Thus, 1X + 9.3* - 163'.