INTRODUCTORY DISCOURSE OF THE 



Mother .id*, TOU find, by mean, of K oom.-trical reasoning respecting this k.nd of triangle, that if squares 

 be drawn on iu three sidea, the large square upon the slanting side opposite the two perpendiculars, is 

 exactlv equal to the smaller squares upon the perpendiculars, taken together; and this is absolutely true, 

 whate'rer be the siie of the triangle, or the proportions of its sides to each other. Therefore, you can 

 alwavs find the length of any one of the three sides by knowing the lengths of the other two. Suppose 

 one perpendicular side to be 8 feet long, the other 4, and you want to know the length of the third aide 

 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.) 



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 perpendicular-sided 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 j the distance 

 of two islands; or the space between the tops of two mountains. 



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 every part of the circle is equally distant 

 from the fixed point or centre. From this fundamental property, an infinite variety of others follow by 

 steps of reasoning more or less numerous, but all necessarily arising one out of another. To give an 

 instance ; it is proved by geometrical reasoning, that if from the two ends of any diameter of the circle 

 you draw two lilies to meet in any one point of the circle whatever, those lines are perpendicular to 

 each other. 



Another property, and a most useful one, is, that the sizes, or areas, of all circles whatever, from the 

 greatest to the smallest, from the sun to a watch-dialplate, are in exact proportion to the squares of 

 their distances from the centre ; that is, the squares of the strings they are drawn with : so that if you 

 draw a circle with a string 5 feet long, and another with a string 10 feet long, the large circle is four times 

 the sire of the small one, as far as the space or area inclosed is concerned ; the square of 10 or 100 

 being four times the square of 5 or 25. But it is also true, that the lengths of the circumferences 

 themselves, the number of feet over which the ends of the strings move, are in proportion to the lengths 

 of the strings ; so that the curve of the large circle is only twice the length of the curve of the lesser. 



But the circle is only one of an infinite variety of curves, all having a regular formation and fixed pro- 

 perties. The oval or ellipte is, perhaps, next to the circle, the most familiar to us, although we more 

 frequently see another curve, the line formed by the motion of bodies thrown forward. When you drop a 

 tone, or throw it straight up, it goes in a straight line ; when you throw it forward, it goes in a curve line 

 till it reaches the ground ; as you see by the figure in which water runs when forced out of a pump, or 

 from a fire-pipe, or from the spout of a kettle or teapot. The line it moves in is called a parabola; every 

 point of which bears a certain fixed relation to a certain poiut within it, as the circle d6es to its centre. 

 Geometry teaches various properties of this curve : for example, if the direction in which the stone is 

 thrown, or the bullet fired, or the water spouted, be half the perpendicular to the ground, that is, half 

 way between being level with the ground and being upright, the curve will come to the ground at a greater 

 distance than if any other direction whatever were given, with the same force. So that, to make the gun 

 carry farthest, or the fire-pipe play to the greatest distance, they must be pointed, not, as you might 

 suppose, level or point blank, but about half way between that direction and the perpendicular. If the 

 air did not resist, and so somewhat disturb the calculation, the direction to give the longest range ought 

 to be exactly half perpendicular. 



The oval, or ellipte, is drawn by taking a string of any certain length, and fixing, not one end as in 



It ta property of nuraben, that every number whatever, whose last place is either 5 or 0, is, when multiplied into itself, 

 qoal to two other which we *|u.re numbers, and divisible by 3 and 4 respectively : thus, 45X45=2025=7294-1296, the squares 

 Of W and 36 ; and 60X60=-3600=1296+2304, the squares of 36 and 48. 



