lar cafes, may be precluded, or in fome measure reftrained.

The positions of geometrical figures are fo various, that it is impoffible to give general rules for the algebraical expreffion of them. The following are a few examples.

4

An angle is expreffed by the ratio of its fine to the radius; a right angle in a triangle, by putting the fquares of the two fides equal to the square of the hypothẹnuse; the pofition of points is afcertained by the perpendiculars from them on lines given in pofition; the pofition of lines by the angles which they make with given lines, or by the perpendiculars on them from given points: The fimilarity of triangles by the proportionality of their fides which gives an equation, &c.

These and other geometrical principles must be employed both in the demonftration of theorems, and in the solution of problems. The geometrical propofition muft firft be expreffed in the algebraical

manner,

1

manner, and the refult after the operation, must be expreffed geometrically.

II. The Demonftration of Theorems.

All propofitions in which the proportions of magnitude only are employed, alfo all propofitions expreffing the relations of the fegments of a ftraight line, of their fquares, rectangles, cubes, and parallelopipeds, are demonstrated algebraically with great ease: Such demonftrations, indeed, may in general be confidered as an abridged notation of what are purely geometrical.

This is particularly the cafe in those propofitions, which may be geometrically deduced without any conftruction of the fquares, rectangles, &c. to which they refer. From the First Proposition of the Second Book of Euclid, the nine following may be cafily derived in this manner, and they may be confidered as proper examples of this moft obvious application of algebra to geometry.

If certain pofitions are either fuppofed or to be inferred in a theorem, we must find, according

E e

1

according to the preceding obfervations, the connection between thefe pofitions and fuch relations of magnitude as can be expreffed and reafoned upon by algebra. The algebraical demonstrations of the 12th and 13th propofitions of the 2d Book of Euclid, require only the 47th of the I. El. The 35th and 36th of the 3d Book require only the 3. III. El. and 47. I. EI.

From few fimple geometrical principles alone, a number of conclufions with regard to figures, may be deduced by algebra; and to this, in a great measure, is owing the extensive use of this fcience in geometry. If other more remote geometrical principles are occafionally introduced, the algebraical calculations may be much abridged. The fame is to be obferved in the fo lution of problems; but fuch in general are lefs obvious, and more properly belong to the ftrict geometrical method.

III. Of the Solution of Problems.

Upon the fame principles are geometrical problems to be refolved. The problem

is

is fuppofed to be conftructed, and proper algebraical notations of the known and unknown magnitudes are to be fought for, by means of which their connections may be expreffed by equations. It may first be remarked, as was done in the cafe of theorems, that in thofe problems which relate to the divifion of the line, and the proportions of its parts, the expreffion of the quantities, and the stating their relations by equations, are fo eafy as not to require any particular directions. But, when various pofitions of geometrical figures, and their properties are introduced, the folution requires more attention and skill. No general rules can be given on this subject, but the following observations may be of use.

I. The conftruction of the problem being fuppofed, it is often farther neceffary to produce fome of the lines till they meet; to draw new lines joining remarkable points; to draw lines from fuch points perpendicular or parallel to other lines, and fuch other operations as feem conducive to the finding of equations; and for this purpose, those

especially

especially are to be employed, which divide the scheme into triangles that are given, right angled or fimilar.

2. It is often convenient to denote by letters, not the quantities particularly fought, but fome others from which they can easily be deduced. The fame may be observed of given quantities.

3. The proper notation being made, the neceffary equations are to be derived by the ufe of the moft fimple geometrical principles, fuch as the addition and subtraction of lines or of fquares, the proportionality of lines, particularly of the fides of fimilar triangles, &c.

4. There must be as many independent equations as there are unknown quantities affumed in the investigation, and from these a final equation may be inferred by the rules of Part I,

If the final equation from the problem be refolved, the roots may often be exhibi

ted