point to the left of A, is a solution of the problem enunciated more generally*.

19. Through a point M equidistant from two straight lines A A' and B B' at right angles to each other, to draw a straight line PMQ, so that the part PQ intersected by A A' and B B' may be of a given length b.

From M draw the perpendicular lines M C, M D.

Let MD = a, D Q = x, CP = y,

then PQ PM + MQ,

or b = √a2 + y2+ √a2 + x2,

a

and = from the similar triangles PC M, M D Q.

α

y

We might solve this recurring equation, and then construct the four roots, as in the last problems; but since the roots of an equation of four dimensions are not easily obtained, we must, in general, endeavour to avoid such an equation, and rather retrace our steps than attempt its solution. Let us consider the problem again, and examine what kind of a result we may expect.

* Lucas de Borgo, who wrote a book on the application of this problem to architecture and polygonal figures, was so delighted with this division of a line, that he called it the Divine Proportion.

Since, in general, four lines P MQ, P' MQ', RSM, R'S' M, may be drawn fulfilling the conditions of the question, the two former, in all cases, though not always the two latter, we may conclude that there will be four solutions; but since the point M is similarly situated with respect to the two lines A A', B B', we may also expect that the resulting lines will be similarly situated with regard to A A and B B'. Thus, if there be one line PM Q, there will be another P' MQ' such that O Q'= OP, and OP'OQ.

Again O S will be equal to O S', and OR to OR'. Hence, if we take the perpendicular from O upon the line SR for the unknown quantity (y), we can have only two different values of this line, one referring to the lines S R and S' R', the other to PQ and P' Q'; hence the resulting equation will be of two dimensions only. In this case the equation is

by2+2a2y - ba2 — 0.

Again, since M R = M R' we may take M H, H being the point of bisection of the line S R, for the unknown quantity, and then also we may expect an equation, either itself of two dimensions, or else reducible to one of that order.

but the square upon R S = square upon R O + square upon

2

So,

X

x +

2

2

an expression of easy construction; the negative value of x is useless: of the remaining two values that with the positive sign is always real, and refers to the lines MSR, MS' R'; the other, when real, gives the lines PMQ, P' M Q'; it is imaginary if b is less than 8a, that is, joining OM and drawing PM Q perpendicular to O M, if b is less than PMQ.

This question is taken from Newton's Universal Arithmetic, and is given by him to show how much the judicious selection of the unknown quantity facilitates the solution of problems. The principal point to be attended to in such questions is, to choose that line for the unknown quantity which must be liable to the least number of variations.

20. Through the point M in the last figure to draw PM Q so that the sum of the squares upon P M and MQ shall be equal to the square upon a given line b.

Making the same substitutions as in the former part of the last article, we shall obtain the equations

x2 + a2 + y2+ a2 = b2, xy=a2,

:. x2 + y2 + 2xy = b2, and x + y =±b,

b

2'

To construct these four values describe a circle with centre M and radius

b

2

cutting A A' in two points L, L'; with centres L, L' and radius describe two other circles cutting A A' again in four points: these are the required points.

21. To find a triangle A B C such that its sides A C, CB, BA, and perpendicular B D, are in continued geometrical progression.

Take any line A B a for one side, let BC= x,

AC: CB CB: BA :: BA: BD;

hence the triangles AC B, A D B, are equiangular, (Eucl. vi. 7, or Geometry, ii. 33,) and the angle ABC is a right angle; also AC=

then

the square upon A C = the square upon B C + the square upon A B;

of these roots two are impossible, since a2 √5 is greater than a2; and of the remaining two the negative one is useless.

B

In A B produced take B Ea√5 (11), and EF =

describe a semicircle, and draw the perpendicular EG; then E G =

22. DETERMINATE problems, although sometimes curious, yet, as they lead to nothing important, are unworthy of much attention. It was, however, to this branch of geometry that algebra was solely applied for some time after its introduction into Europe. Descartes, a celebrated French

philosopher, who lived in the early part of the seventeenth century, was the first to extend the connexion. He applied algebra to the consideration of curved lines, and thus, as it were, invented a new science.

Perhaps the best way of explaining his method will be by taking a simple example. Suppose that it is required to find a point P without a given line AB, so that the sum of the squares on AP and P B shall be equal to the square upon A B.

Let P be the required point, and let fall the perpendicular PM on AB. Let A M= x, M P = y, and A B = a; then by the question, we have

The square on A B = the square on A P + the square on PB. = the squares on AM, M P + the squares on P M, M B,

or a2 = (x2 + y2) + y2 + (a — x)2

= 2 y2 + 2x2

Now this equation admits of an infinite number of solutions, for giving to

or A M any value, such as

a

2

a

3

&c., we may, from the equation,

M

find corresponding values of y or MP, each of them determining a separate point P which satisfies the condition of the problem. Let C, D, E, F, &c., be the points thus determined. The number of the values of y may be increased by taking values of r between those above-mentioned and this to an infinite extent, thus we shall have an infinite number of points C, D, E, F, &c., indefinitely near to each other, so that these points ultimately form a line which geometrically represents the assemblage of all the solutions of the equation. This line A C D E F, whether curved or straight, is called the locus of the equation.

In this manner all indeterminate problems resolve themselves into investigations of loci; and it is this branch of the subject which is by far the most important, and which leads to a boundless field for research*..

23. For the better investigation of loci, equations have been divided into two classes, algebraical and transcendental.

An algebraical equation between two variables r and Y is one which can be reduced to a finite number of terms involving only integral powers of x, y, and constant quantities: and it is called complete when it contains all the possible combinations of the variables together with a constant term, the sum of the indices of these variables in no term exceeding the degree of the equation; thus of the equations

ay + bx + c = o

a y2 + bxy + c x2 + dy + ex + ƒ= 0

the first is a complete equation of the first order, and the next is a complete equation of the second order, and so on.

Those equations which cannot be put into a finite and rational algebraical form with respect to the variables are called transcendental, for

For the definition and examples of Loci, see Geometry, iii. § 6; and the Index, article Locus.

they can only be expanded into an infinite series of terms in which the power of the variable increases without limit, and thus the order of the equation is infinitely great, or transcends all finite orders.

y sin. x, and y = a*, are transcendental equations.

24. The loci of equations are named after their equations, thus the locus of an equation of the first order is a line of the first order; the locus of an equation of the second order is a line of the second order; the locus of a transcendental equation is a transcendental line or curve.

Algebraical equations have not corresponding loci in all cases, for the equation may be such as not to admit of any real values of both x and y; the equation y2 + x2 + a2: O is an example of this kind, where, whatever real value we give to x, we cannot have a real value of y: there is therefore no locus whatever corresponding to such an equation.

THE POSITION OF A POINT IN A PLANE.

Ꮖ

25. The position of a point in a plane is determined by finding its situation relatively to some fixed objects in that plane; for this purpose suppose the plane of the paper to be the given plane, and let us consider as known the intersection A of two lines x X and y Y of unlimited length, and also the angle between them; from any point P, in this plane, draw PM parallel to À Y, and PN parallel to A X, then the position of the point P is evidently known if AM and AN are known. For it may be easily shown, ex absurdo, that there is but one point within the angle Y A X such that its distance from the lines A Y and A X is P N and P M respectively.

AM is called the abscissa of the point P; AN, or its equal MP, is called the ordinate; A M and MP

are together the co-ordinates of P; Xx is called the axis of abscissas, Yy the axis of ordinates. The point A where the axes meet is termed the origin.

The axes are called oblique or rectangular, according as Y A X is an oblique or a right angle. In this treatise rectangular axes as the most simple will generally be employed.

Let the abscissa AM = x, and the ordinate M Py, then if on measuring these lengths AM and MP we find the first equal to a and the second equal to b, we have,

to determine the position of this point P, the two equations

x = a, y = b

P

and as they are sufficient for this object, we call them, when taken together, the equations to this point.

The same point may also be defined by the equation

(y - b)2 + (x a)2 = 0

for this equation can only be satisfied by the values xa and y = b.