LESSONS IN MENSURATION. 



13 





Pig. 3. 



by " ; thiw 23 12' 10* would read twenty-three degree*, twelve 

 uiinutoH, ten seconds. 



v foota will bo well to be remembered by the reader. By 

 Hi,. In: i mi Book of Euclid, it in stated that 



when a ri/lit lino Htunding upon another right line makes the 

 adjacent angles equal to each other, each of these is called a right 

 :i!i-l.'. This condition of two linen U shown in Fig. 2, in which 

 tin) lino AC stands upon tlu> 1m. i> B so as to make the adjacent 

 ii A c, D A c equal ; and in this case each of these angloH 

 is called a right angle. Again, by the 14th Proposition of tbo 

 First Book of Euclid it U shown that if at a point in a ri^nt 

 line two other right linos upon opposite sides of it make the 

 adjacent angles equal together to two right angles, these two 

 lin.-H Hhall be in one ntraight line. So, by reference to Fig. -, 

 if at the point A in tho line A c, the two right lines A B, A D 

 upon opposite sides of it mako the adjacent angles BAG, D A C 

 oqual to two right angles, the linos A B, A D shall bo in ono 

 straight lino ; and it has been assumed that in this figure these 

 adjacent angles are equal to each other, and are equal to two 

 right angles. Therefore, the line D A B is a straight line, and 

 oa it passes through the centre of tho circle B o D it clearly 

 bisects tho circle, that is, it outs it into two equal parts or 

 hemispheres. From this we gather that two right angles 

 together measure the number of degrees 

 contained in half a circle, or j = 180, 

 and hence one right angle measures i* J 

 = 90. 



There is another fact our readers must 

 bear in mind. It has been stated that the 

 arc is the measure of the angle ; but the 

 measurement of this arc in degrees, 

 minutes, etc., is quite irrespective of its 

 size. 



Wo will prove this. Lot A (Fig. 3) be the common centre of 

 the two circles, BCD and B' </ D', of which the circle B' c* D' is 

 double the diameter of tho circle BCD, and let the two straight 

 lines ABB', ADD' be drawn from A to cut the two circles at 

 B B', and D D' respectively. Assume tho angle BAD equal to 

 60, then D B is its measure ; but evidently the angle B' A D' is 

 identical with the angle BAD, and is therefore equal to 60, 

 and B' D' is its measure. Hence, although B' D' is double tho 

 length of B D, it yet measures only the same number of degrees. 

 Once more. Any two angles which together mako up 90 are 

 called complements of one another thus 25 is the complement 

 of 65; and any two angles which together make up 180 are 

 called tho supplements of one another thus 80 is the supple- 

 ment of 100. 



The next step will bring us to the consideration of triangles. 

 This word, derived from the Latin triangulum, implies a figure 

 having three angles, and three sides. It is at once evident 

 that this subject introduces a third element of measurement, 

 namely, surface, or superficies. We have treated of lines, the 

 measure of which is expressed in inches, feet, yards, chains, 

 tc. We have shown how angles are formed by lines, and have 

 explained that tho measure of angles is expressed in degrees, 

 minutes, etc. We now add a third element, namely, surface. 

 So long as only two straight lines were involved, we could in- 



clijde no definite space 

 or surface within them, 

 but the addition of a 

 third line so as to form 

 a triangle at once limits 

 the lengths of the first 

 two, and encloses a 

 space. 



We will first glance 



at the relations which the several lines and angles of a triangle 

 occupy with respect to each other, but must of necessity refer 

 the student to our papers upon Geometry for many introductory 

 points connected with our present subject. 



Euclid has proved, in tho 32nd Proposition of his First 

 Book, that the three interior angles of every triangle are 

 together equal to two right angles, that is, to 180. Let the 

 reader bear this fact in mind. Hence it follows that if tho 

 measure of any two angles of a triangle be known, the third 

 angle can bo found by simple subtraction. For instance, in 

 the triangle ABC (Fig. 4), let tho angle ABC equal 75, and 

 the angle B c A equal 45, the sum of these two angles will be 



Fig. 4. 



Pig. 5. 



Fig. 0. 



75 + 45 = 120 . Then subtract 1 20 from 1 80 (the 

 of two right angles), and the remaining 60 will be the 

 of the angle B A c. 



There are some remarkable fact* in connection with that 

 particular kind of triangle called a right-angled triangle which 

 we will state here, as being calculated to introduce the further 

 consideration of the subject to our readers. 



In his 47th Proposition of the First Book, Euclid has 

 proved tho wonderful (act that in every 

 triangle having one angle a right angle, 

 i.e., 90, the space enclosed by a square 

 constructed upon that side of the triangle 

 opposite the right angle is equal to tho 

 sum of tho two squares constructed upon 

 tho other two sides respectively. 



Lot A B c (Fig. 5) be a right-angled tri- 

 angle, of which B A c is tho right angle. 

 Then a square constructed upon B c will 

 equal in area the squares constructed upon B 

 the two sides AB and AC added together. 

 Tho general formula or expression for this 

 interesting problem U (referring to Fig. 5) B c* = A B* -f- A c*, 

 and therefore BC = VAB J + AC*. In this case we suppose 

 the lengths of A B and A c to be known, and from the above 

 equation B c can bo found. Again, suppose B c and A c to be 

 known, then by transposing the equation, and keeping the un- 

 known side by itself, we have AB 3 =BC 3 AC*; therefore 

 A B = V B c* Ac 2 ; and so by another transposition we can 

 find A c, provided we know the lengths of A B and B c. Hence 

 we arrive at the general and important fact that in every right- 

 angled triangle, if we know the lengths of 

 any two of its sides, we can by simple cal- 

 culation find the third. 



Now this fact can be made use of in 

 a variety of ways. We must first refer 

 the reader to the 4th Proposition of the 

 Sixth Book of Euclid, in which it is stated 

 and proved that in equiangular triangles 

 the sides about the equal angles arc pro- 

 portional. For instance, in Fig. 6 let the 

 two triangles ABC and A 7 B 7 c 7 be equi- 

 angular, the angle A being equal to the 

 angle A 7 , B to B', and c to c 7 ; then A B 

 is to A 7 B 7 as B c is to B 7 c 7 . Again, sup- 

 pose these triangles to be contained, the lesser within the 

 greater, as shown in Fig. 7, and let B' and B be the right 

 angles. Now since the angles A 7 B' c and ABC are both right 

 angles they are equal to each other, and the angle at c ia 

 common to both ; hence the angle B A c is equal to the angle 

 B 7 A 7 c, and the two triangles have the sides about the equal 

 angles proportional, that is, A 7 B 7 is to A B as B 7 c is to B c. But 

 it is the well-known fact of proportion that whenever three 

 quantities are known the fourth can be found. Hence, if A 7 B 7 , 

 B 7 c, and A B be known, the length of B c can be found. 



This rule can be applied to practical use in the following 

 manner : 



Suppose we wish to ascertain the height of a building whose 

 base we can reach. Measure with a chain, or other 

 suitable instrument, a certain distance from the 

 foot of the building. Then at a certain dis- 

 tance from this point, and between it and 

 tho building, erect a perpendicular rod, 

 whose length is known, and let it 

 stand at such a point as that the 

 line of sight between the dis- 

 tance measured and the sum- 

 mit of the building shall 



exactly pass over the c'./ K&* 8. 



top of the rod. Then 

 measure the dis- 

 tance from the 

 bottom of the 

 rod to the 



above-named point, and by the rule of three the height of the 

 building can be ascertained. 



In Fig. 8 let BC represent any building, whose height 

 it is desirable to ascertain. Measure a given distance from 

 B to A, of say fifty feet, then place a rod s'c 7 at a certain 



