NUMERICAL QUESTIONS IN GEOMETRY.

335

word in its largest sense, is the science of measurable quantities, he has to consider both numerical, and what might be called, if the term were not already idiomatically appropriated, metrical proportions. In fact, instruction in geometry presupposes, to a certain extent, the knowledge of number, and involves, throughout, a practical application of that knowledge to the peculiar objects under consideration. Thus, for instance, taking the question, what angles will result from the combination of two straight lines; it is evident that it ought to be divided into two distinct questions, viz.

1. How many angles can be formed with two straight lines?

2. What sorts of angles may, or must they be?

The operation of mind, by which we ascertain, in answer to the first question, that with two straight lines we may form either one, or two, or four angles, is very different from that by which we find, that if we form two or four angles with two straight lines, those angles will be either equal or unequal, which leads at once to the distinction between right angles on one hand, and obtuse and acute angles on the other; and farther, that if two angles formed by two straight lines are unequal, one must necessarily be obtuse, and the other acute; that four angles formed by two straight lines cannot all be unequal, but that two pairs of equal ones must of necessity be formed, and that the position of each angle between the pair from which it differs, is with equal necessity determined; lastly, that if there be only one angle formed by two straight lines, that angle may be either a right, or an obtuse, or an acute angle.

It is again a question of number, to find how many corners are formed in each of the three cases; viz. one, if there be one angle, and none, if there be two or four. But it is a question of measure to ascertain, that the corner which corresponds to the one angle, can, by the greatest possible stretch of that angle, never be reduced to the measure of two right angles; that if the angle be a right one, the measure of the corresponding corner must necessarily be three right angles;

and that if the angle be an acute one, the corner must exceed that measure. Thus, in every geometrical question that can be proposed, a numerical question is inevitably involved, and the clearness with which the pupil shall answer the former, greatly depends upon his having previously solved the latter. It does not, however, follow from this, that the two aspects of the question must always be brought as near together, as they appear in the above example; on the contrary, it will be advisable to let the pupil observe a series of numerical facts, in order that he may collect them under one view, and, if he be capable of it, comprehend them under one general rule, before his attention be at all directed to the geometrical part of the question.

The first general head of exercises, for instance, which ought to be taken up as soon as the child is familiar with the preliminary ideas of point, line, figure, and body, of straight and curve, of perpendicular, horizontal, slanting, &c. is the question: What lines can be drawn between any given number of points? In order to make this question available for instruction, the teacher ought to break it up into a great number of subordinate questions, first with reference to number only, and afterwards with reference to measure. Under given conditions, which by degrees should become more complicated, he ought to present increasing numbers of points, always returning to the question: How many lines can be drawn between so many points so placed? The nature and order of these questions will best be understood from the following table of the answers to which they would lead.

Placing a number of points so, that there shall never be more than two in the same direction, you can draw

1 line between 2 points,

Placing a number of points so, that there shall be three of them in one direction, and the remaining ones never more than two in the same direction, you can draw

Placing a number of points so, that there shall be four of them in one direction, and the remaining ones never more than two in the same direction, you can draw

Placing a number of points so, that there shall be twice. three points in one direction, and the remaining ones never more than two in the same direction, you can can draw 11 lines with 6 points,

Placing a number of points so, that five of them shall form two directions with three points in each, and the remaining ones so, that there shall never be more than two in the same direction, you can draw

It is easy to perceive what a diversity of exercises may thus be deduced from one leading question, not to mention that the various positions in which any given number of points can be placed, form of themselves the subject of a number of interesting preliminary questions. After the teacher has, to a sufficient extent, pursued that part of the subject which refers to the possible number of lines, he should proceed to the question of their respective lengths, in proportion to the distances of their ends. He may, then, first ask: What is the highest number of points that can be placed so, that the lines between them shall all be of equal length?

The pupils having found that three is that number, he may then ask farther: If I have four points, what is the highest number of lines of the same length which I can obtain between them?

If I place them so as to obtain four equal lines between them, will the other two lines be longer or shorter, and what proportion will they bear to each other?

If I place four points so as to obtain three equal lines between them, and at the same time to have three of them in one direction, how many different lines may I obtain, and what will be the proportion of their length? &c.

In the same way questions are to be put respecting the number and extent of intervals and distances which arise out of any number of points. Say, for instance, a number of points be placed so as to have twice four points in the same direction, and the others so, that never more than two points be in the same direction, we shall find with

8 points, 28 distances, 22 intervals, 18 lines,

THEIR RESPECTIVE LENGTH.

339

and, according to the mode of distributing these points, the distances, intervals, and lengths, will bear different relations to each other. For instance, taking the case of nine points, let the four points of one direction be placed at equal distances, the second direction parallel to the first, at a distance from it equal to double the distance of the points on the first, and let the points on the second be twice the distance of those in the first direction; let, lastly, the ninth point be placed so that the distance between it and the two middle points of the first direction be equal to the distance between those two points, and that it be at the same time equidistant from the two middle points of the second direction, thus:

a. Ten sorts of lines, in the following order, beginning from the shortest.

1. be and ce;-2. ge and he;-3. ae and de;-4. ga, gb, hc, and hd;-5. fa, gc, hb, and id;—6. ad;—7. fe, ie, fb, gd, ha, and ic;-8. fc and ib;—9. fd and ia;—10. fi.

b. Nine sorts of intervals, viz.

1. ab, bc, cd, be, and ce;-2. ge and he;-3. ae and de;4. fg, gh, and hi;—5. ga, gb, hc, and hd;—6. fa, gc, hb, and id;-7. fe, ie, fb, gd, ha, and ic;-8. fc and ib;—9. fd and ia. e. Eleven sorts of distances.

1. ah, bc, cd, be, and ce;-2. ge, and he;-3. ae and de;4. fg, gh, hi, ac, and bd;—5. ga, gb, hc, and hd;—6. fa, gc, hb, and id;—7. ad;—8. fe, ie, fb, gd, ha, and ic;-9. fc, ib, fh, and gi;-10. fd and ia;-11. fi.

It will at once be seen that the mere exercise of picking out the various lengths, and the different lines, intervals, and distances, which belong to each length, is in itself calculated to