lar cafes, 'may be precluded, or in some measure restrained. The positions of geometrical figures are so various, that it is impossible to give general rules for the algebraical expression of them. The following are a few examples. An angle is expressed by the ratio of its sine to the radius ; a right angle in a triangle, by putting the squares of the two fides equal to the square of the hypothenuse; the position of points is ascertained by the perpendiculars from them on lines given in position; the position of lines by the angles which they make with given lines, or by the perpendiculars on them from given points : The similarity of triangles by the proportionality of their sides which gives an equation, &c. These and other geometrical principles must be employed both in the demonstration of theorems, and in the solution of problems. The geometrical proposition must firft be expressed in the algebraical manner, and the result after the operation, must be expressed geometrically. manner, II. The Demonstration of Theorems. All propositions in which the proportions of magnitude only are employed, also all propositions expressing the relations of the 'segments of a straight line, of their squares, rectangles, cubes, and parallelopipeds, are demonstrated algebraically with great ease : Such demonstrations, indeed, may in general be considered as an abridged notation of what are purely geometrical. This is particularly the case in those propositions, which may be geometrically deduced without any construction of the squares, rectangles, &c. to which they refer. From the First Proposition of the Sea cond Book of Euclid, the nine following may be easily derived in this manner, and they may be considered as proper examples of this most obvious application of algebra to geometry. If certain positions are either supposed or to be inferred in a theorem, we must find, Ee according 1 according to the preceding observations, the connection between these positions and such relations of magnitude as can be expressed and reasoned upon by algebra. The algebraical demonstrations of the 12th and 13th propositions 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 simple geometrical principles alone, a number of conclusions with regard to figures, may be deduced by algebra ; and to this, in a great measure, is owing the extensive use of this science in geometry. If other more remote geometrical principles are occasionally introduced, the algebraical calculations may be much abridged. The fame is to be observed in the folution of problems; but such in general are less obvious, and more properly belong to the strict geometrical method. III. Of the Solution of Problems. Upon the same principles are geometrical problems to be resolved. The problem: is supposed to be constructed, and proper algebraical notations of the known and unknown magnitudes are to be fought for, by means of which their connections may be expressed by equations. It may first be remarked, as was done in the case of theorems, that in those problems which relate to the division of the line, and the proportions of its parts, the expression of the quantities, and the stating their relations by equations, are so easy as not to require any particular directions. But, when various positions of geometrical figures, and their properties are introduced, the solution requires more attention and skill. No general rules can be given on this subject, but the following observations may be of use. 1. The construction of the problem being supposed, it is often farther necessary to produce some of the lines till they meet; to draw new lines joining remarkable points ; to draw lines from such points perpendicular or parallel to other lines, and such other operations as seem 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 similar. 2. It is often convenient to denote by letters, not the quantities particularly sought, but some others from which they can easily be deduced. The same may be observed of given quantities. 3. proper notation being made, the necessary equations are to be derived by the use of the most simple geometrical principles, such as the addition and subtraction of lines or of squares, the proportionality of lines, particularly of the sides of similar triangles, &c. 4. There must be as many independent equations as there are unknown quantities assumed 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 resolved, the roots may often be exhibi ted |