which the unknown quantity (r) is of the highest power, is made the first, that in which the index of x is less by 1, the second, and so on till the last into which the unknown quantity does not enter, and which is called the absolute term.

Prop. 1. If any number of equations be multiplied, together, an equation will be produced of which the dimension is equal to the sum of the dimensions of the equations multiplied. If any number of simple equations be multiplied together, as aɑ =0, x—b=0, xc=0 &c. the product will be an equation of a dimension containing as many units as there are simple equations. So if higher equations are multiplied together one of a higher order will be produced. Conversely. An equation of any dimension is considered as compound of simple equations or of others the sum of whose dimensions is equal to that of the given one. By the resolution of equations these inferior equations are discovered, and by investigating the component simple equations, the roots of the higher orders are found.

Cor. 1. An equation admits of as many solutions, or has as many roots, as there are simple equations in it.

Cor. 2. Conversely, no equation can have more roots than dimensions.

Cor. 3. Imaginary or impossible roots must enter an equation by pairs, because they arrive from quadratics. And an equation of an even dimension may have all its roots, or an even number of them impossible; but an equation of an odd dimension, must have one positive root at least.

Cor. 4. The roots are positive or negative according to the roots of the simple equations.

Cor. 5. When one root of an equation is discovered one of the simple equations is found; and the given equation being divided by it, will give an equation of a dimension lower by 1.

Prop. II. To explain the general properties of the signs and coefficients of the terms of an equation.

Let r—a=0, x—b=0, x—c=0, the roots are any positive quantities +a, b+, +c, +d, &c. And let x+m=0, x+n=0, &c. be simple equations, whose roots are any negative quantities-m, -n, and let any number of these equations be multiplied together as in the following table:

From this it appears: 1. That in a complete equation the terms are always greater by an unit than the dimension of the equation.

2. The coefficient of the first term is 1. The coefficient of the second term is the sum of the roots with their signs changed. The coefficient of the third term is the sum of all the products made by multiplying any two of the roots together. The coefficient of the fourth term is the sum of all the products that can be made by multiplying together any three of the roots with their signs changed.

The last term is the product of all the roots, with their signs changed.

3. From induction it appears that in any equation there are as many positive roots as there are changes in the signs of the terms, and the remaining roots are negative.

Cor. If a term of an equation be wanting, the positive parts of its coefficient are equal. If there is no absolute term some of the roots =0, and the equation may be depressed by dividing all the terms by the lowest power of the unknown quantity in any of them.

Of the Transformation of Equations.

Prop. I. The affirmative roots of an equation become nega

ALGEBRA.

tive, and the contrary by changing the signs of the alternate terms, beginning with the second.

Thus the roots of the equation x-x3-19x2—49x-30—0, are +1, +2, +3, -5, but the roots of the equation x1+x3—19x2 -49x-30-0, are -1, -2, −3, +5.

Prop. II. An equation may be transformed into another, having its roots greater or less than the roots of the given equation by a given difference.

--

Let e be the given difference, then y=x+e, and x=y+e; and if for and its powers in the given equation y+e and its powers be inserted, a new equation arises whose unknown quantity is y, and its value x+e.

Let the equation proposed be x3-px2+9x-r=0, of which the roots are to be lessened by e. By inserting for a aud its powers y+e and its powers the required equation is

Cor. 1. The use of this transformation is to take away the se cond or any other intermediate erm; for as the coefficients of all the terms of the transformed equation except the first, involve the powers of e, and known quantities only, by putting the coefficient of any term equal to 0, and resolving that equation, a value of e may be determined, which will cause that term to disappear.

Cor. 2. The second term may be taken away by the solution of a simple equation, the third by a quadratic, and so on.

Prop. III. An equation may be transformed into another, whose roots shall be equal to the roots of the given equation multiplied or divided by a given quantity.

e

the new equation will have the property required.

Cor. 1. An equation in which the coefficient of the first term is any unknown quantity, as a, may thus be transformed into another, in which the coefficient of the first termn will disappear. Thus let the equation be ax3-px2+9x—r—0.

y

Suppose yar, or r- and for r and its powers insert ys py2 9y a

a

and its powers, and the equation becomes

=0, or y3-py2+9ay—a3r—0.

Cor. 2. Fractions may be taken away from an equation by multiplying the equation by the denominators, and by this proposition the equation may be then transformed into another without fractions, in which the coefficient of the first term is 1.

Cor. 3. And hence if the coefficient of the second term of a cubic equation be not divisible by 3, the fractions thence arising wanting the second term, may be taken away by the preceding corollary. But the second term may be also taken away, so that there shall be no such fractions in the transformed equation by

supposing x- -,p being the coefficient of the second term.

3

And if the equation ax3—px2+qx-r―0 be given in which p is the transformed equa

not divisible by 3, by supposing x = —

3

За

tion reduced is 23-3p3+9aq ×≈―zp3+9apq-7a'r=0, wanting the second term, having 1 for the coefficient of the first term, and the coefficients of the others all integers.

General Corollary to the preceding Propositions.

If the roots of these transformed equations be found by any method, the roots of the original equation will be found from the simple equations expressing their relation.

BOOKS ON ALGEBRA.

Bonnycastle's Algebra, Fenning's Algebra, to which may succeed the treatises of Frend and Simpson.

PART VII.

GEOMETRY AND MENSURATION.

GEOMETRY is the science and doctrine of local extension, as of

lines, surfaces, and solids, and that of ratios, &c.

The name of geometry literally signifies measuring of the earth, as it was the necessity of measuring the land that first

gave occasion to contemplate the principles and rules of this art, which has since been extended to numberless other speculations; insomuch that, together with arithmetic, geometry forms now the chief foundation of all the mathematics.

Herodotus and Proclus ascribe the invention of geometry to the Egyptians, and assert that the annual inundations of the Nile gave occasion to it; for those waters bearing away the bounds and landmarks of estates and farms, covering the face of the ground uniformly with mud, the people, say they, were obliged every year to distinguish and lay out their lands by the consideration of their figure and quantity; and thus by experience and habit they formed a method or art, which was the origin of geometry.

Geometry is distinguished into theoretical or speculative, and practical.

Theoretical or speculative geometry, treats of the various properties and relations in magnitudes, demonstrating the theorems, &c. And,

Practical geometry is that which applies those speculations and theorems to particular uses in the solution of problems, and in the measurements in the ordinary concerns of life.

Speculative geometry again may be divided into elementary and sublime.

Elementary or common geometry is that which is employed in the consideration of right lines and plane surfaces, with the solids generated from them. And the

Higher or sublime geometry is that which is employed in the consideration of curve lines, conic sections, and the bodies formed of them. This part has been chiefly cultivated by the moderns by help of the improved state of algebra, and the modern analysis or fluxions.

The usefulness of geometry extends to almost every art and science. By its help, astronomers turn their observations to advantage, regulate the duration of times, seasons, years, cycles, and epochas; and measure the distance, motions, and magnitudes of the heavenly bodies. By it geographers determine the figure and magnitude of the earth, and delineate the extent and bearings of kingdoms, provinces, harbours, &c. It is from this science too, that architects derive their just measures in the construction of edifices. It is by the assistance of geometry that engineers conduct all their works, take the situation and plans of towns, the distances of places, and the measure of such things as are only accessible to the sight. It is not only an introduction to fortification, but highly necessary to most mechanics, especially carpenters, joiners, mathematical instrument-makers, and all who profess designing. On geometry like vise depends the theory of