PROJECTION OF MATHEMATICAL DIAGRAMS. The dia grams by which mathematical students (and even writers) represent their solid figures aro generally so imperfect, that it may be worth while to explain how, in all Cases of sufficient importance, a good drawing may be made with very little trouble. The demonstration may be found in the' Cambridge Mathematical Journal,' No. 8, p. 92. The projection is supposed to be the OteTnoonarmc, in which the eye is at an infinite distance, and all parallels are projected into parallels, &c.
Let o x, o v, 0 z, be the intended projections of the three axes of co-ordinates, the dark lines being supposed to belong to that quarter of space in which lies a line drawn to the eye from the origin o. Each of the angles Y o z, z o x, x 0 Y, is then greater than a right angle. The following table contains numbers sufficiently near for the purpose, proportional to the square roots of the sines of twice the angles written in the opposite columns.
0 , o o The lines between the degrees are to be understood as representinE half degrees : thus opposite to 991 should come 129r —1•104*. The use of this table is ss follows :—suppose the angles Y o z, z 0 X, x o Y to be severally 104i*, and 1324" : thus :— Opposite to the angles put down the numbers belonging to them it the table, and opposite to each number the co-ordinate whose capital letter does not appear in the angle. Then opposite to x, y, and z, we
have 956, 696, and 998. These numbers show the proportions which the projections of equal lines bear to one another on the three axes, Thus a foot parallel to x is to a foot parallel to y, as 956 to 696 in the projection. If then a card be taken, and the angle z o x be cut out ; and if a slit be made in the direction of o Y, just wide enough to permit a pencil to travel, scales of equal parts may be laid down on o x, OY, and o z, which shall represent the projections of equal lines in the three directions ; and this may be done once for all. It would be easy enough to make a general scale by which the equal parts proper for any angle should be taken out at once.
The isometrical perspective of Professor Farish [PERSPECTIVE, col. 416] is the simplest case of this, namely, that in which the angles are each 120'. The only difference between this particular case and any other is, that the former requires only one scale of equal parts, whereas the latter requires either two or three. In other respects this method of using them is precisely the same.