Objects, Advantages , And Pleasures Of Science.

upon by common arithmetic * And questions infinitely more difficult can easily be solved by the rules of algebra . In like manner , by arithmetic you can tell the properties of particular numbers ; as , for instance , that the number 348 is divided by three exactlyso as to leave

, nothing over ; but algebra teaches us that it is only one of an infinite variety of numbers , all divisble b y 3 , and any one of which you can tell the moment you see it ; for they all have the remarkable property , that if you add together the

figures they consist ^ of , the sum total is divisible by 3 . You can easily perceive this in any one case , as in the number mentioned , for 3 added to 4 and that to 8 make 15 , which is plainly divisible by 3 ; and if you divide 348 by 3 , you find the quotient to be 116 , with nothing over . But this does not at all prove that any

number , the sum of whose figures is divisible by 3 , will itself also be found divisible by 3 , as 741 ; for you must actually perform the division here , and in every other case , before you can know that it leaves nothing over . Algebra , on the

contrary , both enables you to discover such general properties , and to prove them in all their generality . ' ! 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 threesided 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 opposide 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 ruus 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 sideand 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 liue 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