slat pokłos pinos 7v Axirews. 16. Kupov, toy twv lepowy | angle, and therefore a body will travel down it with a greater βασιλεα, επι τη τε αρετη και τη σοφια θαυμαζομεν. velocity, and it is found that this increase of speed exactly makes up for the greater distance. EXERCISE 34.- ENGLISH-GREEK. The other fact is, that if a body has to fall from one point 1. The flocks follow the shepherd. 2. The king has care of to another not in the same vertical line, (for) the citizen. 3. Ears are tired by the idle talk of the old as, for instance, from D to E, the line woman. 4. An old woman is talkative. 5. The shepherd leads of quickest descent is not along the the herd of oxen to the city. 6. Oxen are sacrificed to the gods straight line joining these two points, by (úto with gen.) the priests. 7. O priests, sacrifice an ox but along some curve, as D FE. The to the gods. 8. Children love their (the) parents. 9. Parents reason of this is, that if the body be are loved by their children. 10. It is the business of a good moving down the curve it will at any shepherd to take (have) care of his herds. moment be at a lower level than it would if falling down the incline DE; KEY TO EXERCISES IN LESSONS IN GREEK.-IX, and since the velocity of a falling body, EXERCISE 25.-GREEK-ENGLISH. as we have seen, depends upon the ver1. The ravens croak. 2. Avoid flatterers. 3. Keep away from the tical distance passed over, its velocity Fig. 100. deceiver. 4. Men delight in the harp, in the dance, and in song. 5. is all along greater. The space passed Horses are driven by whips. 6. The harps delight the minds of men, over is, however, greater too; but this is more than compensated 7. A grasshopper is friendly to a grasshopper, and an ant to an ant. for by the increased velocity. The curve of shortest descent of all 8. The shepherds sing to their pipes. 9. Among the Athenians there is found to be that which has the greatest curvature without were contests between quails and cocks. 10. The shepherds drive the he rising as it approaches E. If a pencil be fixed so as to project flocks of goats into the meadows. 11. The life of ants and quails is horizontally from the rim of a wheel, and made to trace a curte very laborious. 12. Many have a good countenance, but a bad voice. on paper while the wheel is rolling on, it will be exactly that of EXERCISE 26.-ENGLISH-GREEK. shortest descent. As we shall see further on, there are other 1, DEVYW Kolaka, 2. Kopares kpwcovoi. 3. Tepregte popuryyt. 4. OoxnOnor remarkable and important properties possessed by this curve, Tous avOpwmovs TEPOVOT. 5. Elavrovow in Tous Maoriyye. 6. oi truoc twv which is called a Cycloid. (See Lessons in Geometry, XXIII., ανθρωπων ελαύνονται φορμιγγι. 7. Αί συριγγες τερπονσι τους ποιμενας. 8. page 309.) Αί αιγες εις τον λειμωνα ελαύνονται. 9. Ο ποιμην αδει προς την συριγγα. 10. PROJECTILES. Καλην μεν ωπα εχει η θυγατηρ, κακην δε οπα. Having thus seen the laws which govern the motion of falling EXERCISE 27.-GREEK-ENGLISH. bodies, we pass on naturally to notice the movements of pro1. The birds sing. 2. Favour begets favour, (and) strife (begets) jectiles. Here, of course, as before, the resistance of the air strife. 3. We count youth happy. 4. Need begets strife. 5. Rich impedes motion to a greater or less extent. This resistance men often conceal their baseness by (means of) wealth. 6. O fair boy, increases as the square of the velocity, for if the speed of a love your good brother and your fair sister. 5. Avarice is the mother body be doubled, it not only has to displace twice the bulk of seness. 8. The poor are often happy. 9. Wisdom air, but it must do it with twice the velocity, and for this a fourin the hearts of men stirs up marvellous longings for the beautiful. fold force is needed. As, however, our calculations would be 10. Death sets men free from their cares. 11. Friendship springs much complicated if we took this into consideration, we will up by means of resemblance (in disposition). 12. Wine creates neglect it, but we must remember to make allowance for it in laughter. 13. Deliberation comes to the wise in the night. 14. The our results. wise punish baseness. 15. Men often delight themselves with light The path of a projectile is in a curve called a parabola, that (or vain) hopes. EXERCISE 28.-ENGLISH-GREEK, is, a curve similar to the one which we should obtain if we were to cut a cone in a direction parallel to one side. (See Les1. Opvides qdovol. 2. Xapır xapıv TIKTEV, epis epiv. 3. Lopuq eyeupetai ev τοις των ανθρωπων θυμοις θαυμαστος ,ερως αγαθων 4. Τερπομαι ωδη των sons in Geometry, XXI., page 251.) We can, however, trace this path in a simpler way. ορνιθων. 5. Αί ωδαι των ορνιθων τερπουσι τον ποιμενα. 6. Τερπομεθα ορνισι. 7. Oi avpwrot ETOVTQI Tois avake. 8. Oi av@putou reovrat ty avakti. When a body is projected with any velocity, as, for example, when a bullet is fired from a gun, it is acted apon by two forces EXERCISE 29.-GREEK-ENGLISH. -the original velocity with which it was started, which, 29 we are 1. In difficult matters few companions are faithful. 2. The suppliants not considering the resistance of the air, we may consider to be touch our knees. 3. Death is a separation of the soul and body. 4. a uniform force; and, secondly, the attraction of the earth, which Wealth furnishes men with various aids. 5. Do not yield to the words is an accelerating force, causing it to fall 16 feet in the first of wicked men. 6. Do not, my son, be a slave to the service of the second, 48 in the next, and so on. Now from a knowledge of body. 7. The Greeks pour cups of milk as libation-offerings to the these two motions we can easily tell at what point the body will nymphs. 8. Accustom yourself to, and exercise your body with, toil and sweat. 9. Chatterers vex (or weary) the ear with repetitions (of be at any given moment; the same story). 10. Accustom your soul, my son, to good deeds. 11. | and by thus finding seveEvil stories do not lay hold of our ears. 12. We listen with our ears. ral different points in its 13. Do not hate a friend for a small fault. 14. My son, taste the milk. course we can trace out 15. The soldiers bear lances. its path. Let the bullet be pro- jected from the point a 1. Ω νεανιαι, εθιζετε τα σωματα συν πονω και ιδρωτι. 2 Ορεγομεθα των | (Fig. 101), in the direcαγαθων πραγμάτων, 3. Πολλοι τερπονται χρυσω. 4. Εξ αγαθον πρώματος γιγνεται κλεος. 5. Τους καλους μυθους των σοφων θαυμαζομεν. 6. Τα των velocity, and take ac of αγαθων ανθρωπων αγαθα πραγματα θαυμαζεται. 7. Οι στρατιωται μαχονται such a length as to re- present the space it would Draw A D vertically D downwards to represent 16 feet, and complete the LAWS OF FALLING BODIES--PROJECTILES-COLLISION Fig. 101. AC and A D represent, then, the two forces acting on the ballet; 7 are two remarkable facts that have been discovered in and, since each produces its full effect, it will at the end of one on with the laws of bodies falling down an incline that second have arrived at the point E. Since, however, the foree of just notice here. The first is, that if we take any gravity is not uniform, the line A E, which represents its path, will of chords, A E, B E, eto. (Fig. 100), all meeting in E, the not be straight, but curved upwards, for when a half of AC ha: Joint of the circle, and make inclined planes parallel been passed over, gravity will only have caused it to move overs oportional in length to them, a body will take the same quarter of AD. If now we draw through x a straight line EF. wfall down each of these inclines. BE, for instance, is parallel to A B and equal to Ac, it will represent the motion of .ch longer ian DE, yet it is inclined at a much greater the bullet from its original impulse during the next secord. To represent gravity we must take E G, three times the length of AD, Now raise c to the fourth division, and let it fall against D. and thus, by completing the parallelogram, we find that at the No momentum will be destroyed; it will merely be shared end of this second the bullet has arrived at h. In the same way, between the two balls, as much being gained by the one as is lost by making H L equal to five times AD, we find K to be the point by the other; and, since both balls have the same weight, each at which the bullet will have arrived after three seconds, and in will move with half the velocity that c had on striking D. They this way we can map out its whole path. will therefore rise toWe see from this the reason why the sights of a rifle are gether to the first divi.. arranged as they are. If the bullet travelled in a perfectly sion of the arc DF, for straight line, the soldier would aim directly at the point hec takes twice as long (E wished to hit; but the force of gravity acts on the bullet, and to fall from 4 as it therefore he has to point the rifle at & point as much above it does from 1, and the veas the bullet will fall in the time it takes to travel the distance. locity is proportional If, for instance, it takes two seconds for the ball to reach the to the time, therefore -target, he must aim at a point 64 feet above it. To do this it acquires a double would be very inconvenient and uncertain, as he would be velocity in falling. anable to tell whether the point he was aiming at was directly Now whatever veloover the mark. The sight at the end next the stock is therefore city a body acquires made to adjust to different elevations above the barrel, according in falling from any Fig. 102. to the distance of the object aimed at; and thus, though the height, it must start rifleman sees the two sights in a straight line with it, the with that velocity to rise to that height. A velocity, then, half barrel is really pointed considerably upwards, as will be evident as great as that acquired by c will raise the two balls to 1. to & bystander. In the same way, if we make c half as heavy as D, and raising There is one other fact relating to projectiles, which, though it to the 9th division let it fall, the two will, as before, rise to 1. It seems strange, is a necessary result of the second law of The mass moved after impact is three times that of c, the motion. velocity will therefore be only one-third as great; they will If a body be projected horizontally, no matter how great its therefore rise the height. We see thus that when one body velocity be, it will always reach the earth in exactly the same strikes against another, the momentum will be divided between time as if it fell vertically. The speed in falling is not in any them, and hence the resulting velocity will be as much less than way interfered with by the horizontal motion. that of the moving body as the mass of the two is greater than its mass. COLLISION OR IMPACT. For example, suppose a ball weighing 1 lb. and moving with a We said that any force is measured by the velocity generated velocity of 60, to impinge against a larger ball weighing 14 lbs. in a second. There is one class of forces, however, which cannot The mass after impact will be 15 lbs., or fifteen times that of the be so measured, because they do not act for any appreciable ball; the velocity will therefore be is, or 4 feet per second. No length of time. These we call impulses or impulsive forces; momentum is lost. The original momentum was 1 x 60; after. any force which is of the nature of a blow is placed in this impact it is 15 x 4, which is also equal to 60. class. This principle supplies us with a means of measuring very When one body strikes against another different results will great velocities, as that of a cannon-ball or other missile. ensne, according to the nature of the bodies. If an ivory ball A large block of wood or metal is suspended by a rod so as to be allowed to fall on a stone slab, it rebounds or rises from its swing to and fro with as little friction as possible. This is surface, but the height to which it rises is less than that from called a ballistic pendulum. Against this the ball is caused to which it fell. Were the ball perfectly elastic, it would rise to strike, and by its impact it sets it in motion. A graduated arc the same height. This, however, is not all that has occurred, is fixed under the block on which the distance to which it the changes have been more complicated. On striking the slab, swings can be noted, and from this we can calculate the velocity the ball is first flattened in a slight degree. In proof of this we it had immediately after the ball struck it. We have only to may emear the slab with oil, and we shall find the ball marked, measure the vertical height to which it rose, and ascertain the not in a minute point as it would be if merely laid on it, but velocity it would attain in falling from that height, and thas over a space increasing in size with the violence of the blow. we have the velocity with which it started. The particles are thus compressed, but their own elasticity | The weight of the bullet and the pendulum being also known, causes them at once to recover their original position, and in we can at once determine the proportion they bear to each other, 80 doing the ball flies up from the slab. and thus we can ascertain the velocity of the ball from that of The effect, then, varies with the degree of elasticity of the the pendulum. body. We can, however, only consider the cases of elastic and Suppose, for example, that the pendulum weighs half a ton, inelastic bodies, not that any substances are perfectly so, but and being struck by a ball weighing 24 lbs. is raised to a height by examining these we shall get at general principles, which can of 16 feet. In falling from this height it would acquire a then be applied or modified as may be required. velocity of 32; this, therefore, is that which it had immediately We will first consider the case of inelastic bodies, and well. after the ball struck it. But the mass of the ball is to that of kneaded clay or patty may be chosen as suitable substances to the two together as 24 to 1144, or 1 to 48 nearly. The velocity esperiment with. Wax, softened with oil, will also answer well. of the ball was therefore 48 x 32, or 1536 feet per second. In making experiments on impact, the best plan is to procure Hence we see why, if one body strikes against another, the balls of the substances chosen, and, having fastened them to heavier it is as compared with that against which it strikes, the strings, suspend them in such a way that they may just touch greater the effect produced. If we want to drive a large nail or one another. to strike a violent blow, we use a heavy hammer, for by it we Let us take two such balls, c and D (Fig. 102), of equal obtain a much greater momentum, and thus accomplish the weight, and having raised them to the same height, in opposite work with greater ease. So, too, when we are driving a nail directions, leave them free to fall together and strike each other. | into a plank, we place a support behind or hold a heavy hammer Since both fall from the same height, their velocities are equal, against it. Unless we do this the momentum is si and they each have the same mass; their momenta are there board, which yields to the blow, and thus destroys much of the fore equal, and being in opposite directions neutralise each other. | effect. But when a heavy inelastic body is held behind, this, Both balls will therefore, after impact, remain at rest. too, has to share the momentum, and thus the plank yields In order to measure the distance through which the balls fall, much less, and the nail is driven more easily. we must draw the arcs which they describe and divide them. In the same way some of the feats of strength sometimes We do not, however, make the divisions equal, but draw a series exhibited may be explained. A man will lie with his shoulders of parallel lines at the same distance apart, the lowest being supported on one chair and his feet on a second. A heavy anvil even with the top of the balls, and make our divisions at the is then placed on his body, and on this he allows stones to be points where these cut the arcs. The reason of this is, that the broken or blows to be struck, which, but for the anvil Velocity is proportional, not to the length of arc, but to the certainly kill him. The reason is, that the moment: Tertical height, and thus these divisions indicate the velocity. hammer imparts but a very slight velocity to the 40. Is is evident, since a concrete quantity can only be compared with another of the same kind (Obs. 11, Lesson XXVII., Vol. II., - I deo be s siastic page 102), that the fourth quantity determined must be of the QUE : . : the same kind as the third quantity. In order that the ratios of ya L o ugh the, the two pairs of quantities may be equal, either two must be - ... - - amenal Laws, of one kind and two of another, or all four must be of the same 2017 . - S U V un together, kind. 2. Suppose we have the following question proposed :. . 3. ami allow it to fall EXAMPLE.-If the rent of 40 acres of land be £95, what will 11. Des but the momentum be the rent of 37 acres ? N e rastic, they will be It is evident that the sam required must bear the same ratio VERI" . 2 uning their shape, being | to £95 that 37 acres do to 40 acres. tu. . . posms , w.. altroy the motion of candl Hence we have, writing the ratios in the form of fractions, W erefore remain at rest. and Sum required. – 37 acres – ***. .: m ell o chat which chad. If a £95 40 acres = the abstract number 0. " xiad so as just to touch one Therefore the sum required = !! * £95, which can be reduced . V to pounds, shillings, and pence. Since 40 acres cost £95, 1 acre costs £ ; And therefore 37 acres cost pounds. if two balls be allowed to fall. w " Sun. E ben und. We see, then, that no 4. In solving such a question by the Rule of Three, the state. We we n t to sell or than it was in the case of in. / ment of the proportion is generally written thus : and sh e was '*** Nared between all the balls, as it acres, acres. Xem ide , med et stod periments cun, of course, be varied 40 : 37 : : £95 : sum required. w o wote w a ar re ommended to try them for Then, by equating the product of the extremes and means, we wametan in Bad Wys learnt by seeing or trying a few ex get the result. We have put the first example, however, in e are no meaning out wany. As, however, there is the fractional form, in order to indicate clearly the fact that Ako strong an d supplung ivory balls, the experiments the ratio of the two quantities of the same kind (acres in this b or in de weg op way with common glass marbles. Lay case) is an abstract number, by which the other quantity, the Horia and women who kamooth surface, like the top of a £95, is multiplied. When we state the question in the second bisa dikoniy v a 413** so that a marble may just roll way, and talk about multiplying the means and extremes web t ashlandi wi botter till, ont a sinall groove in which together, some confusion might arise from the idea of multido the ja A Y naar uw marble may then be laid in the plying 37 acres by £95. The fact to be borne in mind is that W e want dan wil to mtriko it gently. The latter will the rule is merely the expression of the fact that the ratios of m ind all the all, while the other will move. The reason two pairs of quantities are equal. in der man who oulutely to rest is, that glass is not! 5. The example we have given is what is called a case of pa ty elastio, and thus the reaction is not direct Proportion--that is to say, if one quantity were increased, ong una, to strike the end, the same re | pounds thoy cost would be increased. Th is one other law relating to impact. sponding quantities be increased, the other is proportionally 1114 that the angle of incidence is equal to the diminished, the case is one of what is called Inverse Proportion. bino All'ull filleotion." The meaning of this will For instance : look for froin the annexed figure. Let any EXAMPLE.--If 35 men eat a certain quantity of bread in 20 Hide darkesa hakim 140 AC, in the direction D B, it will re- | days, how long will it take 50 men to eat it ? Helen in de the content D, making tho same angle with Here, evidently, the more men there are, the less time will T ural otan Hill I door. The angle DDF, or that they take to eat the bread; hence, as the number of men all I had it 141, 1 illod the angle of in idence, while increases, the corresponding quantity of the other kind-viz., The H ill b ion, and the law a-sarts that these the number of days-decreases. akimirin pipe to optics and other branches of Hence, since 50 men are more than 35 men, the required Timoniall Hullullrillustrations of this law. number of days will be fewer than the 20 days which corte spond to the 35 men. wirin XAMPLES IN LESSON XVII. In stating the proportion, therefore, in order to make the i o delle t re feet, and will reach the enrth again in ratios equal, if we place the larger of the two terms of one ratio in the first place, we must place the larger of the two terms of imali 14, whileli equals 30! 16, or 484 feet. the other ratio in the third place. I tilh with a velocity of 160. Thus, placing 50 men in the first place, we must put 20 l itt i wila, iu the last of which it will fall 208 fset. days (which, we can see, will be larger than the required answer) in the third place, and then the statement would be II y s uda, and pass over 576 feet. correctly made thus : 50 : 35 : : 20 days : required number of days. ARONA IN ARITTIMETIC.--XXXII. Therefore the required number of days = 20 * days = 14 days. N.B.-We might reduce the example to a case of Direct MUN THRI# HINUL AND DOUBLE. Proportion thus, which will, perhaps, explain the above method a tha application of the principles of more clearly :para quantition. We have shown 35 men eat of the bread in one day. 1 page 1119) that if any three numI alwaya be found much that the four 50 " required number of days » male, llence, il throe concrete Hence, since the quantity eaten in one day will increase with u hiinli arg of the anmo kind, and the the number of men, we have A finally gjuntity of the same kind as As 35 : 50 :: med that it will ber tho samo ratio 6 : required number of days; # He drop two biti tu cach other ; Therefore required time = 20 ~ 2. = 14 days, as before. 20 x 35 1 days in 1 6. The last question might also have been solved thus : GEOMETRICAL PERSPECTIVE.—III. 35 men eat the quantity in 20 days; BEFORE proceeding farther and deeper into our subject, we wish Therefore 1 man eats ss of the quantity in 20 days; to draw the pupil's attention to an explanation of projection, a Therefore 1 man eats a ton of the quantity in 1 day; term applied not only to perspective but also to other systems of 35 x 20 50 representation, namely, orthographic and isometric. Our reason Therefore 50 men eat 20.235 of the quantity in 1 day; for introducing this now, is in order to make it clearly under 20 x 35 And therefore 50 men would occupy 50 , or stood how the plan of an object is to be treated when we are eating the bread. 20 x 35 about to make a perspective drawing of that object, as we very frequently meet with cases when the plan of the object to be N.B.—To get the time occupied in doing a certain work, represented must be drawn according to the position which that when the amount done per unit of time (say per day) is given, object presents, whether horizontal or inclined. The plan, as we must evidently divide the whole quantity of work by the we said in Lesson I., is produced by perpendicular lines drawn amount done in a day. In the case given above, the bread from every part of an object upon a horizontal plane. Now, being considered the unit, quas of the bread is eaten in 1 day, | there can be no difficulty in drawing a plan when the subject represented by it is parallel with the ground or horizontal and therefore 50, which is the whole amount eaten divided plane; but it occurs sometimes that it is placed at an angle 20 x 35 with both planes, that is, with the picture-plane and groundby the amount eaten in one day, will be the whole time occupied. | plane: therefore in cases of this kind it is necessary to under stand the first principles of orthographic projection, namely, projec7. Hence we get the following statement of tion by straight lines upon vertical and horizontal planes. We Simple or Single Rule of Three. have mentioned above another method of projection, isometric; Write down the ratio of the two quantities which are of the as the term has been introduced, we will explain its meaning same kind, putting the greater in the first place. Then ob and then pass it by, as it does not, like orthographic, form any serving from the nature of the question whether the fourth auxiliary to perspective. The term isometric signifies likequantity required will be greater or less than the third one measurement, that is, all the parts of the drawing, both near which is given, place tho greater of the two in the third place and distant, are drawn to one and the same scale, also the plan of the proportion, and multiply the extremes and means and elevation are combined in one drawing. It is a method together. much used by architects and engineers when they wish to give what is generally called a bird's-eye view of a building, etc., EXERCISE 51.-EXAMPLES IN SINGLE RULE OF THREE. without diminishing the distant parts, as shown in perspective 1. If 16 barrels of flour cost £28, what will 129 cost? projection. A drawing made isometrically will enable a stranger 2. I1 641 sheep cost £485 15s., what will 75 cost ? to understand the proportions, position, and general character 3. If £11 5s. buy 63 pounds of tea, how many can be bought for of a subject probably better than any other system; hence the £385 ? 4. A bankrupt pays 6s. 4d, in the pound: what will be received on reason of its frequent use. a debt of £2,563 10s.? | The extent to which we intend to proceed with orthographic 5. What is £1,460 worth in dollars, allowing 4 dollars 84 cents to projection must be limited to that which relates to, and can a pound, and 100 cents to a dollar ? assist us in, our present subject, by which we hope to make it 6. If lb. of snuff cost £f, what will 150 lbs. cost ? a valuable auxiliary in our efforts to render the science of per7. A man bought of a vessel, and sold ; of what he bought spective easy and intelligible. for £8,240, which was just the cost of it: what was the The difference between the results of perspective and ortho. whole vessel worth? graphio projection is caused by the altered position of the eye 8. If, of a yard cost of a crown, what will 37 yards cost ? 9. If 10 men build a wall in 7 days, how long would it take 24 when viewing the object. In perspective the eye is in one place men to build it? only, and from that place is included all that can be seen within 10. If 6 men build a wall in 15 days, how many men would it take the angle of sight. In orthographic projection the eye is supjust to finish it in 22 days? posed to be opposite every part at the same time, above the object 11. It of a ton costs 9s. 810., what would 41 of a cwt. cost? when the plan is represented, and before it when the elevation is 12. If a twopenny loaf weighs 1 lb. 2 oz. when wheat is 50s. a represented; consequently, in perspective, all the visual rays quarter. what should it weigh when wheat sells for 60s. ? proceeding from the object to the eye converge to one point; 13. If the weight of a cubic inch of distilled water be 25373'grains, but in orthographic projection these rays are drawn parallel and a cubic foot of water weighs 1000 oz, avoirdupois, find the with each other, and perpendicularly to the plane of projection, number of grains in a pound avoirdupois. 14. If 1 lb. avoirdupois weighs 7,000 grains, and 1 lb. troy weighs whether the plane is horizontal or vertical. To make this clear, 5,760 grains, find how many pounds avoirdupois are equal to we request the pupil to compare Figs. 5 and 6 of the last 175 lbs. troy. Lesson with Fig. 8, when he will notice that the characteristic 15. Find the rent of 27a. 3r. 15p. at £1 3s. 6d. per acre. difference between the two systems rests entirely upon the 16. The price of standard silver being 5s. 6d. per ounce, how many different treatment of the lines of projection, which, as we have shillings are coined out of a pound troy ? said, converge in one case, and are parallel in the other. Fig. 8 17. A bankrupt's assets are £1,500 10s., and he pays Is. 3;d. in the | is to show how a cube is projected orthographically upon vertical pound : what are his debts ? and horizontal planes of projection. A is the vertical, and B the 18. If standard gold is worth 1}!!d. per grain, how many sovereigns horizontal. c is the cube in space, that is, at a distance from would be coined out of a pound troy of gold ? 19. What is the income of a man who pays 53s. 100. tax when it | both planes of projection. If straight lines are drawn from the 18 7d. in the pound? angles of the cube perpendicularly to and meeting the plane B, 20. Raising the income tax 1d. in the pound increases my am and then lines (a, b, c, d) be drawn to unite them, we shall have of tax by £2 3s. 4d., and the tax I actually pay is £15 3s. 4d. : a plan of the cube; and as the edges in this case are placed what is the rate of the income-tax ? perpendicularly with the ground, the plan will be a square. 21. A barrel of beer lasts a man and his wife 3 weeks, she drinking Again, if horizontal and parallel lines are drawn from the half the amount he does : how long would it last 5 such angles of the cube until they meet the vertical plane A, and are couples ? then joined by the lines e, f, g, h, we shall produce the elevation; and because the horizontal edges of the cube are perpendicular KEY TO EXERCISE 50, LESSON XXXI. (Vol. II., page 270). to the vertical plane of projection, the drawing in this case also will be a square. Consequently, it will be seen that the drawing £ 8. d. £ 8. d. 15. •025. of the plan or the elevation is the same size as the object on the 8. 45 12 6. 16. "115. 2. 5 6 51. 9. respective plane to which the object is parallel, according to the 48 5 & 17. *889. 14 103. 10. 91 This result 5 0. given scale of that object, as in Figs. 10 and 11. -938. 8 114. 11. 132 6 3. 19. 3 785. makes orthographic projection of much importance for practical 5. 35 17 1. 12. 157 15 CA. 15.997. purposes. The working drawings for the guidanco cf builders 6. 13. 01. and mechanists are made by this method. Horizontal longths 14. 023. and breadths are shown both in the plan and eleration, but 13. 4. |