12 



THE POPULAR EDUCATOR. 



KEY TO EXERCISES IN LESSONS IN GREEK. XXXII. 

 EXERCISE 94. GREEK-ENGLISH. 



1. I was setting upright. 2. I was playing drunken pranks. 3. I 

 jnade a disturbance. 4. I have set upright. 5. I was serving. 6. I 

 was living. 7. I was supporting. 8. I was narrating. 9. I have 

 built. 10. I was throwing. 11. I was leading. 12. I have hoped. 

 13. I have entreated. 14. I have associated. 15. I have lamented. 

 16. I was praying. 17. I spent. 18. I was following, 19. I had 

 founded. 20. I had taken. 21. I have been dug. 22. I was casting 

 away. 23. I was preparing. 24. I was in a state of displeasure. 25. 

 I have been a benefactor. 26. I have narrated. 



KEY TO EXERCISES IN LESSONS IN GREEK. XXXIII. 

 EXERCISE 95. GREEK-ENGLISH. 



1. The soldiers were ordered to go against the enemy. 2. Sparta 

 was once fearfully shaken by an earthquake. 3. The power of the 

 Persians has been broken by the Greeks. 4. The enemy were shut up 

 in the citadel. 5. The barbarians took to flight when they heard the 

 Greeks dash their shields against their spears. 6. The war was 

 stopped. 7. We hope that we shall accomplish all things well. 8. 

 I would that I might accomplish all things well. 9. The treaty has 

 been broken by the barbarians. 



EXERCISE 96. ENGLISH-GREEK. 



1. Ol <rrpari<a-rai irpot rout Tro\ffttovt i 

 *H jroXiv ijLteTCpa iiiro aeicr^iou Teffpavfna 

 6paw?0n<reTai. 4. 'H TToXit vvo atiapov 

 VTTO TWV 'EXXrji/wi/ tQpoivaOri. 6. Oi iroXef 

 ciotv. 7. A! ao-Tridet irpot ^a. Sopa-ra vtro 



opevf<r9at KeK6\tv<rftevoi eio-iv. 2. 

 . 3. EKCivrj n TroXi? wjro <TCtfffjiOV 

 eietat. 5. 'H TO)I/ Tlepffiav <W,"<v 

 ioi ett rr)v aKpav KaTaK6K/\ei<r;ui/Oi 

 ^lav iroXe/ouwi/ (KpovaOnaav. 8. 'O 



Xe^iot TrejrauTai. 9. 'O TroXejUor TrfiravcrfTai 

 tfjiftta. 11. KeXeuirai pa&iov ecrnv n avva 



10. E<0e waina KaXa>r 

 12. 'H trvv0riKn viro -ri 



LESSONS IN MENSURATION. I. 



MENSURATION is a comprehensive and general term, signifying 

 the determination of the extent both of lines, surfaces, and 

 solids, and is derived from the Latin word mensura, a measure ; 

 and it is our purpose to explain in the following chapters, as 

 simply as possible, the rules by which the science is governed. 



In our treatment of Geometry (which is, after all, but a branch 

 of Mensuration) we have explained what are the relations, pro- 

 portions, and properties of lines and surfaces. Under the head 

 of Mensuration we shall show the mode of estimating the 

 lengths, surfaces, and capacities formed by lines and angles. 

 And herein lies the difference between the two subjects, for 

 whilst Geometry simply treats of the general relations of lines 

 and angles, Mensuration enters into the methods for determining 

 their length and extent in individual cases. 



In order to avoid repetition, we will refer our readers to our 

 chapters upon Geometry for the definitions which are necessary 

 to be understood in studying the subject of Mensuration. 



It will strike every person upon reflection that all measure- 

 ments must be included under four distinct heads : the first, of 

 lines ; the second, of angles, that is, of the inclination of two 

 lines to each other which meet ; the third, of surfaces, that is, 

 of spaces included or shut in by lines, but devoid of thickness ; 

 and the fourth, of solids, that is, of bodies possessed of length, 

 breadth, and thickness. Everything possessed of magnitude 

 can be classed under one or other of these four distinct heads, 

 and we propose to adopt the order in which we have stated 

 them in our consideration of measurements generally. 



And first as to lines. The measurement of lines, which at 

 first sight appears a very simple process, is by no means so easy 

 a matter as it appears. We are, of course, speaking not of 

 approximate, but of correct measurement. It is by no means 

 easy to ensure perfect uniformity undevialing equality in the 

 length even of the self-same thing. The dimensions of all 

 bodies are affected in a greater or less degree by differences of 

 temperature, and however minute this difference may be, yet 

 when the body or instrument so affected is intended to be used 

 as a standard or guide wherewith to measure other and longer 

 "ines, an error, however trifling, becomes speedily doubled, 

 tripled, and so on, until it has grown serious. 



Our national standards of measurement are on this account 

 most scrupulously protected, and if required for reference must 

 be used with the greatest caution, particularly as regards 

 temperature. 



It is not, however, necessary in the ordinary routine of busi- 



ness to be so minutely exact as, for instance, to bring a powerful 

 microscope to bear upon the point where the rod, rule, or chain 

 has to repeat itself in order to secure perfect coincidence at the 

 point of meeting. Indeed, in the use of that valuable measurer 

 of length, the Gunter chain an instrument wo shall have 

 again to refer to a man accustomed to the work will bring its 

 back extremity so nearly to coincide with the point where the 

 front end of the chain last touched, that after many hundred 

 repetitions of the operation, a second measurement by calcula- 

 tion will detect but a few inches of difference. 



In measurements of length, when the distance to be measured 

 is trifling, recourse is had to a foot rule, a yard measure, or a 

 ten-foot rod ; but in longer distances, the measurement of land 

 for instance, the " Gunter chain " is employed, for reasons which 

 will be explained when we come to treat of land surveying. 

 This chain consists of 100 iron links united by iron rings. The 

 full length of the chain is 66 feet, consequently each link and 

 its accompanying ring is ^ of a foot in length, or 7'92 inches. 

 Every ten links from either end is distinguished by a brass 

 label having one or more notches cut in it, the number of 

 notches corresponding to the number of tens from the end 

 nearest it, and the middle or fiftieth link having a circular piece 

 of brass attached to it. These marks are intended to save time 

 and trouble in counting the number of any particular link from 

 the extremity of the chain. 



Another point for consideration in measuring accurately a. 

 long line is to guard against any deviation from its intended 

 route. If it be a straight line, the course throughout must be 

 absolutely straight, and to accomplish this it will be necessary 

 either to fix upon a given landmark of small lateral dimensions 

 which lies exactly in the intended line, and to direct each suc- 

 cessive extension of the chain upon this point by the eye, from 

 the back end of the chain, or previously to stake out by means- 

 of rods the line of route, and to be careful that the chain lies 

 always evenly along that line. In the measurement of a curved 

 line, the rods employed to stake it out must stand sufficiently 

 close together as that an almost inappreciable difference shall 

 exist between the straight lines which connect them and the 

 curve of which they form a part. Correctness in the measure- 

 ment of lines is absolutely essential to correctness in the mea- 

 surement of the spaces enclosed by them ; this 

 fact cannot be too carefully borne in rnind. 



Our next step is the consideration and 

 measurement of the inclination of two straight 

 lines to each other which meet, that is, of 

 tho angle formed by their meeting or inter- 

 section. Mensuration in this respect is 

 simply the application of arithmetic to trigo- 

 nometry. We shall not at present go 

 deeply into the subject of trigonometry, but 

 merely explain the rules upon which the measurement of angle;? 

 is based. 



It is proved by geometry that the angles at the centre of a 

 circle bear to one another the same proportion as the arcs, oi' 

 portions of the circumference of the circle which the lines form- 

 ing the angles cut off from it. 



In Fig. 1, let B c D be a circle of which A is the centre, and 

 let the line A B be drawn, and suppose it fixed. It is evident 

 that, as from the centre of a circle any number of lines or radii 

 can be drawn from the centre to the circumference, we can draw 

 A c, A c' in any position we please, and thus form any number 

 of angles B A c, B A c' at the point A. Now the measure of 

 these angles is estimated, not by the lines which form them, as 

 A B, A c, A c', but by the arcs of the circle these lines cut off ; 

 thus, the measure of the angle B A c is the 

 arc B c, and so on. It is, therefore, only 

 necessary to adopt some method of dividing 

 these arcs in order to measure arithmetically 

 the angles they represent. 



Now it has been decided that every com- 

 plete circle shall be considered as divisible 

 into 360 equal parts, each of these parts to be 

 called a degree ; again, each degree shall be 

 divisible into sixty equal parts, called minutes; 

 minute into sixty equal parts, called seconds. 



Fig. 2. 



and each 

 The division 



can be carried further, but it is not usual to extend it beyond 

 seconds. The signs by which these several divisions are 

 recognised are : A degree, by ? ; a minute, by '; and a second, 



