Art.

523. The axis of z one directrix, and the thread of a screw the other

Page

524. A straight line passes through two straight lines and a given curve, to find the surface

252

253

525. A straight line, parallel to a given plane, passes through two given curves, to find the surface 526. A straight line passes through three given straight lines, to find the surface . 253

CHAPTER IX.

ON CURVES OF DOUBLE CURVATURE.

527-9. The meaning of the term curve of double curvature

530. The curve arising from the intersection of a sphere and cylinder 531. The curve arising from the intersection of a cone and Paraboloid 532. Surfaces found on which a curve of double curvature may be traced. 533-5. To find out when the intersections of surfaces are plane curves 536. To find the curve represented by the equations

538. To describe a Curve of double curvature by points. Examples

ALGEBRAICAL GEOMETRY.

PART I.

APPLICATION OF ALGEBRA TO PLANE GEOMETRY.

CHAPTER I.

INTRODUCTION.

1. THE object of the present Treatise is the Investigation of Geometrical Theorems and Problems by means of Algebra.

Soon after the introduction of algebra into Europe, many problems in plane geometry were solved by putting letters for straight lines, and then working the questions algebraically; this process, although of use, did not much extend the boundaries of geometry, for each problem, as heretofore, required its own peculiar method of solution, and therefore could give but little aid towards the investigation of other questions.

It is to Descartes that we owe the first general application of algebra to geometry, and, in consequence, the first real progress in modern mathematical knowledge; in the discussion of a problem of considerable antiquity, and which admitted of an infinite number of solutions, he employed two variable quantities x and y for certain unknown lines, and then showed that the resulting equation, involving both these quantities, belonged to a series of points of which these variable quantities were the co-ordinates, that is, belonged to a curve, the assemblage of all the solutions, and hence called "the Locus of the Equation."

It is not necessary to enter into further details here, much less to point out the immense advantages of the system thus founded. However, in the course of this work we shall have many opportunities of explaining the method of Descartes; and we hope that the following pages will, in some degree, exhibit the advantages of his system.

2. In applying algebra to geometry, it is obvious that we must understand the sense in which algebraical symbols are used.

In speaking of a yard or a foot, we have only an idea of these lengths by comparing them with some known length; this known or standard length is called a unit. The unit may be any length whatever: thus, if it is an inch, a foot is considered as the sum of twelve of these units, and may therefore be represented by the number 12; if the unit is a yard, a mile may be represented by the number 1760.

But any straight line A B fig. (1) may be taken to represent the unit of length, and if another straight line CD contains the line A B an exact number (a) of times, CD is equal to (a) linear units, and omitting the words "linear units," C D is equal to (a).

B

If C D does not contain A B an exact number of times, they may have a common measure E, fig. (2); let, then, CD m times Em E, and A Bn E, then C D has to A B the same ratio that m E has to n E, or

m

m

that m has to n, or that has to unity; hence C D = times A B

m

n

n

= b.

If the lines A B and C D have no common measure, we must recur to considerations analogous to those upon which the theory of incommensurable quantities in arithmetic is founded.

We cannot express a number like 2 by integers or fractions consisting of commensurable quantities, but we have a distinct idea of the magnitude expressed by 2, since we can at once tell whether it be greater or less than any proposed magnitude expressed by common quantities; and we can use the symbol 2 in calculation, by means of reasoning founded on its being a limit to which we can approach, as nearly as we please, by common quantities.

Now suppose E to be a line contained an exact number of times in A B, fig. (2), but not an exact number of times in C D, and take m a whole number, such that m E is less than C D, and (m+1) E greater than C D. Then the smaller E is, the nearer m E and (m + 1) E will be to C D; because the former falls short of, and the latter exceeds, C D, by a quantity less than E. Also E may be made as small as we please; for if any line measure A B, its half, its quarter, and so on, ad infinitum, will measure A B. Hence we may consider CD as a quantity which, though not expressible precisely by means of any unit which is a measure of A B, may be approached as nearly as we please by such expressions. Hence C D is a limit between quantities commensurable with E, exactly as 2 is a limit between quantities commensurable with unity.

We conclude, then, that any line C D may be represented by some one of the letters a, b, c, &c., these letters themselves being the representatives of numbers either integral, fractional, or incommensurable.

3. If upon the linear unit we describe a square, that figure is called the square unit.

Let CDFE, fig. (1), be a rectangle, having the side CD containing (a) linear units CM, MN, &c., and the side C E containing (b) linear units C O, O P, &c., divide the rectangle into square units by drawing lines parallel to CE through the points M, N, &c., and to CD through the points O, P, &c. Then in the upper row COQD there are (a) square units, in the second row OPRQ the same, and there are as many rows as there are units in C E, therefore altogether there are (b×a) square units in the figure, that is, CF contains (a b) square units, or

is equal in magnitude to (ab) square units; suppressing the words

66

square units," the rectangle C F is equal to a b.

If C D 5 feet and C E = 3 feet, the area CF contains 15 square

feet.

The above proof applies only to cases where the two lines containing the rectangle can be exactly measured by a common linear unit.

Suppose C D to be measurable by any linear unit, but CE (fig. 2) not to be commensurable with CD; then, as has been shown, we may find lines CM, CN commensurable with CD approaching in magnitude as nearly as we please to C E.

Completing the rectangles CP and C Q, we see, that as C M and CN approach to CE, the rectangles CP and CQ approach to the rectangle CF, that is, the rectangle CE, CD is the limit of the rectangle C M, CD, just as CE is the limit of CM. Let therefore a and b be respectively the commensurable numbers representing C D and C M, and let c be the incommensurable number expressing CE, then the rectangle CE, CD the limit of the rectangle C M, M P the limit of the number a b, by the first part of this article, the product of the respective limits of a and b a c.*

=

=

=

Hence, generally, the algebraical representative of the area of a rectangle is equal to the product of those of two of its adjacent sides.

If ba, the figure CF becomes the square upon CD, hence the square upon C D is equal to (a × a) times the square unit = a2.

We are now able to represent all plane rectilineal figures, for such figures can be resolved into triangles, and the area of a triangle is equal to half the rectangle on the same base, and between the same parallel lines.

4. To represent a solid figure, it will be sufficient to show how a solid rectangular parallelopiped may be represented.

Let a, b, c, be, respectively, the number of linear units in the three adjacent edges of the parallelopiped; then, dividing the solid by planes parallel to its sides, we may prove, as in the last article, that the number of solid units in the figure is a xbx c, and, consequently, the parallelopiped equal to axbxc.

The proof might be extended to the case where the edges of the parallelopiped are fractional, or incommensurable with the linear unit.

Ifbca, the solid becomes a cube, and is equal to a×a×a, or a3. 5. We proceed, conversely, to explain the sense in which algebraic expressions may be interpreted consistently with the preceding observations.

*That "the product of the limits of two incommensurable numbers is the limit of their product," may be thus shown. Let v and w be incommensurable numbers, and let = m + m' and w = n + n', m and n being commensurable numbers, and m' and a diminishable without limit; that is, and w are the respective limits of m and n, then vw = mn + mn + nm' + m'n', the right-hand side of this equation ultimately becomes mn, and the left-hand side of the equation is the product of the limits.

Algebraic expressions may be classed most simply under the form of homogeneous equations, as follows:

In the first place, each equation may be understood as referring to linear units; thus, if L be put instead of the words the linear units,' the equations may be written

x times L a times L,

x2 times L + ax times L, or (r2+ ax) times L = bc times L,

(x2 + ax3 + bcx2 + defx) times L = ghkl times L, and so on. The solution of each equation gives a times L in terms of (a, b, c, . . .) times L; and thus the letters a, b, c, x are merely numbers, having reference to lines, but not to figures.

This will be equally true if L is not expressed, but understood; and it is in this sense that we shall interpret all equations beyond those of the third order.

The same reasoning would equally apply if we assumed L to represent the square or cubic unit, only it would lead to confusion in the algebraic representation of a line.

6. Again, these equations may, to a certain extent, have an additional interpretation.

For if we consider the letters in each term to be the representatives of lines drawn perpendicular to each other, the second equation refers to areas, and then signifies that the sum of two particular rectangles is equal to a third rectangle; the third equation refers to solid figures, and signifies, that the sum of three parallelopipeds is equal to a fourth solid.

Moreover we can pass from an equation referring to areas to another referring to lines, without any violation of principle; for, considering the second equation as referring to areas, the rectangles can be exhibited in the form of squares; and if the squares upon two lines be equal, the lines themselves are equal, or the equation is true for linear units.

7. It follows as a consequence of the additional interpretation, that every equation of the second and third order will refer to some geometrical theorem, respecting plane or solid figures; for example, the second equation, when in the form 2a (ax) is the representation of the well-known problem of the division of a line into extreme and mean ratio.

By omitting the second and third terms of the third equation, and giving the values of 2a, a, and a to d, e and f, respectively, we obtain the algebraic representation of the ancient problem of the duplication of the cube.

8. The solution of equations leads to various values of the unknown quantity, and there are then two methods of exhibiting these values; first, by giving to a, b, c, &c., their numerical values, and then performing any operation indicated by the algebraic symbols.