10 OBJECTS, ADVANTAGES, AND 



7,543,233 by itself, and that product again by the original number, you 

 would have to multiply a number of seven places of figures by an equally 

 large number, and then a number of 14 places of figures by one of seven 

 places, till at last you had a product of 21 places of figures a very 

 tedious operation ; but working by logarithms, you would only have to 

 take three times the logarithm of the original number, and that gives 

 the logarithm of the last product of 21 places of figures, without any 

 further multiplication. So much for the time and trouble saved, which 

 is still greater in questions of division ; but by means of logarithms 

 many questions can be worked, and of the most important kind, which 

 no time or labour would otherwise enable us to solve. 



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 respect- 

 ing this kind of triangle, that if squares be drawn on its three sides, 

 the large square upon the slanting side opposite the two perpendicu- 

 lars, is exactly equal to the two smaller squares upon the perpendicu- 

 lars, taken together ; and this is absolutely true, whatever be the size 

 of the triangle, or the proportions of its sides to each other. There- 

 fore, 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 10 feet long, the other 6, and you want to know the length 

 of the third side opposite to the perpendiculars, you have only to 

 find a number such, that if multiplied by itself, it shall be equal to 10 

 times 10, together with 6 times 6, that is 136. (This number is be- 

 tween 11^ and 11-f-.) Now only observe the great advantage of know- 

 ing 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 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 mea- 

 suring the inclination 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 ends of the measured lines 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 



