rithms many questions can be worked, and of the most important kind, which no time or labour would otherwise enable us to resolve.

Geometry teaches the properties of figure, or particular portions of space, and distances of points from each other. Thus, when you see a triangle, or three-sided figure, one of whose sides is perpendicular to another side, you find, by means of geometrical reasoning respecting this kind of triangle, that if squares be drawn on its three sides, the large square upon the slanting side opposite the two perpendiculars, is exactly equal to the two smaller squares upon the perpendiculars, taken together; and this is absolutely true, whatever be the size of the triangle, or the proportions of its sides to each other. Therefore, you can always find the length of any one of the three sides by knowing the lengths of the other two. Suppose one perpendicular side to be 3 feet long, the other 4, and you want to know the length of the third side opposite to the

perpendicular; you have only to find a number such, that if, multiplied by itself, it shall be equal to 3 times 3, together with 4 times 4, that is 25* (This number is 5.)

Fig. 1.

Fig. 2.

3

4

[Fig. 1. is the 47th Proposition, without the lines by which it is demonstrated. The square of 5 is equal to the two squares of 304 taken together.]

[Fig. 2. represents the extension of the same Proposition to other figures; the two semicircles, or circles, or ovals, on the two perpendicular sides being together equal to the semicircle, circle, or oval, on the opposite side.]

* It is a property of numbers, that every number whatever, whose last place is either 5 or 0, is, when multiplied into itself, equal to two others which are square numbers, and divisible by 3 and 4 respectively:—thus, 45 × 45=2025=729+1296, the squares of 27 and 36; and 60 × 60=3600=1296+2304, the squares of 36 and 48.

Now only observe the great advantage of knowing this property of the triangle, or of perpendicular lines. If you want to measure a line passing over ground which you cannot reach-to know, for instance, the length of one side covered with water of a field, or the distance of one point on a lake or bay from another point on the opposite side-you can easily find it by measuring two lines perpendicular to one another on the dry land, and running through the two points; for the line wished to be measured, and which runs through the water, is the third side of a perpendicularsided triangle, the other two sides of which are ascertained. But there are other properties of triangles, which enable us to know the length of two sides of any triangle, whether it has perpendicular sides or not, by measuring one side, and also measuring the inclinations of the other two sides to this side, or what is called the two angles made by those sides

with the measured side. Therefore you can easily find the perpendicular line drawn, or supposed to be drawn, from the top of a mountain through it to the bottom, that is the height of the mountain ; for you can measure a line on level ground, and also the inclination of two lines, supposing them drawn in the air, and reaching from the two ends of the measured line to the mountain's top; and having thus found the length of the one of those lines next the mountain, and its inclination to the ground, you can at once find the perpendicular, though you cannot possibly get near it. In the same way, by measuring lines and angles on the ground, and near, you can find the length of lines at a great distance, and which you cannot approach: for instance, the length and breadth of a field on the opposite side of a lake or sea; the distance of two islands; or the space between the tops of two mountains.

[The base, or ground line AB being measured, and the angles at A and B, made by AF and BF, AE and BE, geometry enables us to calculate the length of the lines BF and BE, and from thence the length of FC and ED, the plumb-lines supposed to be dropt from the tops of the two mountains and the line EF, that is, to ascertain the heights of the mountains, and the distance of their tops from each other.]

Again, there are curve-lined figures as well as straight, and geometry teaches the properties of these also. The best known of all the curves is the circle, or a figure made by drawing a string round one end which is fixed, and marking where its other end traces, so that