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very slowly. PELITARIUS, a French mathematician, in a treatise which bears the date of 1558, is the first who observed that the root of an equation is a divisor of the last term; and he remarked also this curious property of numbers, that the sum of the cubes of the natural numbers is the square of the sum of the numbers themselves.
Vieta was a very learned man, and an excellent mathematician, remarkable both for industry and invention. He was the first who employed letters to denote the known as well as the unknown quantities, so that it was with him that the language of algebra first became capable of expressing general truths, and attained to that extension which has since rendered it such a powerful instrument of investigation. He also gave new demonstrations of the rule for resolving cubic, and even biquadratic equations. He also discovered the relation between the roots of an equation of any degree, and the coefficients of its terms, though only in the case where none of the terms are wanting, and where all the roots are real or positive.
About the same period, Algebra became greatly indebted to ALBERT GIRRARD, a Flemish mathematician, whose principal work, Invention Nouvelle en Algebre, was printed in 1669. This ingenious author perceived a greater extent, but not yet the whole of the truth, partially discovered by Vieta, viz. the successive formation of the coefficients of an equation from the sum of the roots; the sum of their products taken two and two; the same taken three and three, &c. whether the roots be positive or negative. He appears also to have been the first who understood the use of negative roots in the solution of geometrical problems, and is the author of the figurative expression, which gives to negative quantities the name of quantities less than nothing; a phrase that has been severely censured by those who forget that there are correct ideas, which correct language can hardly be made to express. The same mathematician conceived the notion of imaginary roots, and showed that the number of the roots of an equation could not exceed the exponent of the highest power of the unknown quantity. He was also in possession of the very refined and difficult rule, which forms the sums of the powers of the roots of an equation from the coefficients of its terms.
By HARRIOT, the method of extracting the roots of equations was greatly improved; the smaller letters of the alphabet, instead of the capital letters employed by Vieta, were introduced.
The succession of discoveries, above related, brought the algebraic analysis, abstractly considered, into a state of perfection, little short of that which it has attained at the present moment. It was thus prepared for the step which was about to be taken by Descartes : this was the application of the algebraic analysis, to define the nature, and investigate the properties, of curve lines, and, consequently, to represent the notion of variable quantity.
This author begins with the consideration of such geometrical
problems as may be resolved by circles, and straight lines; and a. plains the method of constructing algebraic formulæ, or of translating a truth from the language of algebra into that of geometry. He then proceeds to the consideration of the problem, known among the ancients by the name of the locus ad quatuor reclas, and treated of by Apollonius and Pappus. The algebraic analysis afforded a method of resolving this problem in its full extent; and the consideration of it is again resumed in the second book. The thing required is, to find the locus of a point, from which, if perpendiculars be drawn to four lines given in position, a given function of these perpendiculars, in which the variable quantities are only of two dimensions, shall be always of the same magnitude. Descartes shows the locus, on this hypothesis, to be always a conic section; and he distinguishes the cases in which it is a circle, an ellipsis, a parabola, or a hyperbola. It was an instance of the most extensive investigation which had yet been undertaken in geometry, though, to render it a complete solution of the problem, much more detail was doubtless necessary. The investigation is extended to the cases where the function, which remains the same, is of three, four, or five dimensions, and where the locus is a line of a higher order, though it may, in certain circumstances, become a conic section. The lines given in position may be more than four, or than any given nnmber; and the lines drawn to them may either be perpendiculars, or lines making given angles with them.
In this book also, an ingenious method of drawing tangents to curves is proposed by Descartes, as following from his general principles. Fermat was far more fortunate with regard to this problem, and his method of drawing tangents to curves, is the same in effect that has been followed by all the geometers since his time,
The leading principles of algebra were now unfolded, and the notation was brought, from a mere contrivance for abridging common language, to a system of symbolical writing, admirably fitted to assist the mind in the exercise of thought. The happy idea, indeed, of expressing quantity, and the operations on quantity, by conventional symbols, instead of representing the first by real magnitudes, and enunciating the second in words, could not but make a great change on the nature of mathematical investigation. The language of mathematics, whatever may be its form, must always consist of two parts; the one denoting quantities simply, and the other denoting the manner in which the quantities are combined, or the operations understood to be performed on them. GEOMETRY expresses the first of these by real magnitudes, or by what may
be called natural signs; a line by a line, an angle by an angle, an area by an area, &c.; and it describes the latter by words. ALGEBRA, on the other hand, denotes both quantity, and the operations on quantity, by the same system of conventional or arbitrary symbols. Thus, in the expression si-ax+63 the letters a, b, c, denote quantities but the ierras x?, a 1', &c. denote certain operations performed on those quantities, as well as the quantities themselves; r is the quantity raised to the cube; and ar? the same quantity. x raised to the square, and then multiplied into a, &c.; the combination, by addition or subtraction, being also expressed by the signs + and
Now, it is when applied to this latter purpose that the algebraic language possesses such exclusive excellence. The mere magnitudes themselves might be represented by figures, as in geometry, as well as in any way whatever; but the operations they are to be subjected to, if described in words, must be set before the mind slowly, and in succession, so that the impression is weakened, and the clear apprehension rendered difficult. In the algebraic expression, on the other hand, so much meaning is concentrated into a narrow space, and the impression made by all the parts is so simultaneous, that nothing can be more favourable to the exertion of the reasoning powers, to the continuance of their action, and their security against error. Another advantage resulting from the use of the same notation, consists in the reduction of all the different relations among quantities to the simplest of those relations, that of equality, and the expression of it by equations. This gives a great facility of generalization, and of comparing quantities with one another. A third arises from the substitution of the arithmetical operations of multiplication and division, for the geometrical method of the composition and resolution of ratios. Of the first of these, the idea is so clear, arrd the work so simple; of the second, the idea is comparatively so obscure, and the process so complex, that the substitution of the former for the latter could not but be accompanied with great advantage. This is, indeed, what constitutes the great difference in practice between the algebraic and the geometric method of treating qnantity. When the quantities are of a complex nature, so as to go beyond what in algebra is called the third power, the geometrical expression is so circuitous and involved, that it renders the reasoning most laborious and intricate. The great facility of generalization in algebra, of deducing one thing from another, and of adapting the analysis to every kind of research, whether the quantities be constant or variable, finite or infinite, depends on this principle more than any
other. Few of the early algebraists seem to have been aware of these advantages.
The use of the signs plus and minus has given rise to some dispute. These signs were at first used the one to denote addition, the other subtraction, and for a long time were applied to no other purpose. But as, in the multiplication of a quantity, consisting of parts connected by those signs, into another quantity similarly composed, it was always found, and could be universally demonstrated, that, in uniting the particular products of which the total was made up, those of which both the factors had the sign minus before them, must be added into one sum with those of which all the factors had the sign plus ; while those of which one of the factors had the sign plus, and the other the sign minus, must be subtracted from the same, this general rule came to be more simply expressed by saying, in multiplication like signs gave plus, and that unlike signs gave minus.
Hence the signs plus and minus were considered, not as merely denoting the relation of one quantity to another placed before it, but, by a kind of fiction, they were considered as denoting qualities inherent in the quantities to the names of which they were prefixed. Even the most scrupulous purist in mathematical language must admit, that no real error is ever introduced by employing the signs in this most abstract sense. If the equation r3 +pro +qr-r=0, be said to have one positive and two negative roots, this is certainly as exceptionable an application of the term negative, as any that can be proposed; yet, in reality, it means nothing but this intelligible and simple truth, that If + prqr-=(x-2)(r+b)(x+c); or that the former of these quantities is produced by the multiplication of the three binomial factors, -a, +b, I+c. We might say the same nearly as to imaginary roots; they shew that the simple factors cannot be found, but that he quadratic factors may be found; and they also point out the means of discovering them. The aptitude of these same signs to denote contrariety of position among geometric magnitudes, makes the foregoing application of them infinitely more extensive and more indispensable.
In the end of the sixteenth century, the time and labour consumed in astronomical and other calculations had become excessive, and were felt as extremely burdensome by the mathematicians and astronomers all over Europe. Napier of Merchiston, whose mind seems o have been peculiarly turned to arithmetical researches, and who vas also devoted to the study of astronomy, had early sought for he means of relieving himself and others from this difficulty. He nad viewed the subject in a variety of lights, and a number of ingenous devices had occurred to him, by which the tediousness of arithnetical operations might, more or less completely, be avoided. In the course of these attempts, he did not fail to observe, that whenever the numbers to be multiplied or divided were terms of a geometrical progression, the product or the quotient must also be a term of that progression, and must occupy a place in it pointed out by the places of the given numbere, so that it might be found from mere inspection, if the progression were far enough continued. If, for instance, the third term of the progression were to be multiplied by the seventh, the procluct must be the tenth, and if the twelfth were to be divided by the fourth, the quotient must be the eighth; so that the multiplication and division of such terms was reduced to the addition and subtraction of the numbers which indicated their places in the progression. It is plain, however, that the resource of the geometrical progression was sufficient, when the given numbers were terms of that progression ; but if they were not, it did not seem that any advantage could be derived from it. Napier, however, perceived, and it was by no means obvious, that all numbers whatsoever might be inserted in the progression, and have their
places assigned in it. After conceiving the possibility of this, the next difficulty was, to discover the principle, and to execute the arithmetical process, by which these places were to be ascertained.
The way in which he satisfied himself that all numbers might be intercalated between the terms of the given progression, and by which he found the places they must occupy, was founded on a most ingenious supposition,—that of two points describing two different lines, the one with a constant velocity, and the other with a velocity always increasing in the ratio of the space the point had already gone over: the first of these would generate magnitudes in arithmetical, and the second magnitudes in geometrical progression; and it is plain, that all numbers whatsoever would find their places among the magnitudes so generated. The numbers which indicate the places of the terms of the geometrical progression, are called by Napier the logarithms of those terms. Various systems of logarithms, it is evident, may be constructed according to the geometrical progression assumed; and of these, that which was first contrived by Napier, though the simplest, and the foundation of the rest, was not so convenient for the purposes of calculation, as one which soon afterwards occurred, both to himself and his friend Briggs, by whom the actual calculation was performed.
The first writer on the subject of Mechanics is ARCHIMEDES. He treated of the lever, and of the centre of gravity, and has shown that there will be an equilibrium between two heavy bodies connected by an inflexible rod or lever, when the point in which the lever is supported is so placed between the bodies, that their distances from it are inversely as their weights. The same great geometer gave a beginning to the science of Hydrostaties, and discovered the law wh ch determines the loss of weight sustained by a body on being immersed in water, or in any other fluid. Archimedes, therefore, is the person who first made the application of mathematics to natural philosophy.
The mechanical inquiries, begun by the geometer of Syracuse, were extended by Ctesibius and HERO; by ANTHEMIUS of Tralles; and, lastly, by PAPPUS ALEXANDRINUS. Ctesibius and Hero were the first who analyzed mechanical engines, reducing them all to combinations of five simple mechanical contrivances, to which they gave the name of Aurapers, or Powers, the same which they retain at the present moment. Even in mechanics, however, the success of these investigations was limited ; and failed in those cases where the reso lution of forces is necessary, that principle being then entirely unknown.
Galileo was born at Pisa in the year 1564, and as early as 1592 published a treatise, della Scienza Mechanica, in which he gave the theory, not of the lever only, but of the inclined plane and the screw; and also laid down this general proposition, that mechanical engines make a small force equivalent to a great one, by making the former move over a greater space in the same time than the latter,