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Πατροκλος φιλος ην Αχιλλέως, 16. Κύρον, τον TWV Пepσwv angle, and therefore a body will travel down it with a greater βασιλέα, επί τῇ τε αρετη και τη σοφία θαυμαζόμεν.

EXERCISE 34.-ENGLISH-GREEK.

1. The flocks follow the shepherd. 2. The king has care of (for) the citizen. 3. Ears are tired by the idle talk of the old woman. 4. An old woman is talkative. 5. The shepherd leads the herd of oxen to the city. 6. Oxen are sacrificed to the gods by (no with gen.) the priests. 7. O priests, sacrifice an ox to the gods. 8. Children love their (the) parents. 9. Parents are loved by their children. 10. It is the business of a good shepherd to take (have) care of his herds.

KEY TO EXERCISES IN LESSONS IN GREEK.—IX.
EXERCISE 25.-GREEK-ENGLISH.

1. The ravens croak. 2. Avoid flatterers. 3. Keep away from the deceiver. 4. Men delight in the harp, in the dance, and in song. 5. Horses are driven by whips. 6. The harps delight the minds of men. 7. A grasshopper is friendly to a grasshopper, and an ant to an ant. 8. The shepherds sing to their pipes. 9. Among the Athenians there

were contests between quails and cocks. 10. The shepherds drive the flocks of goats into the meadows. 11. The life of ants and quails is very laborious. 12. Many have a good countenance, but a bad voice. EXERCISE 26.-ENGLISH-GREEK.

1. Φευγω κολακα, 2. Κορακες κρώζουσι. 3. Τερπεσθε φόρμιγγι. 4. Ορχηθμοι τους ανθρώπους τέρπουσι. 5. Ελαύνουσιν ίππους μαστιγγι. 6. Οἱ θυμοι των ανθρώπων ελαύνονται φόρμιγγι. 7. Αἱ συριγγες τέρπουσι τους ποιμένας, 8. Αἱ αίγες εις τον λειμώνα ελαυνονται. 9. Ὁ ποιμην φάει προς την σύριγγα. 10. Καλήν μεν ωπα έχει ή θυγατηρ, κακην δε οπα.

EXERCISE 27.-GREEK-ENGLISH.

1. The birds sing. 2. Favour begets favour, (and) strife (begets)

strife.

3. We count youth happy. 4. Need begets strife. 5. Rich men often conceal their baseness by (means of) wealth. 6. O fair boy, love your good brother and your fair sister. 7. Avarice is the mother of every kind of baseness. 8. The poor are often happy. 9. Wisdom in the hearts of men stirs up marvellous longings for the beautiful. 10. Death sets men free from their cares. 11. Friendship springs up by means of resemblance (in disposition). 12. Wine creates laughter. 13. Deliberation comes to the wise in the night. 14. The wise punish baseness. 15. Men often delight themselves with light (or vain) hopes. EXERCISE 28.-ENGLISH-GREEK.

1. Ορνιθες άδουσι. 2. Χαρις χαριν τίκτει, ερις εριν. 3. Σοφια εγείρεται εν τους των ανθρώπων θύμοις θαυμαστος έρως αγαθών. 4. Τέρπομαι ωδη των ορνιθών. 5. Αἱ ωδαι των ορνίθων τέρπουσι τον ποιμένα. 6. Τερπόμεθα όρνισι. 7. Οι άνθρωποι έπονται τοις αναξι. 8. Οι ανθρωποι πείθονται την ανακτί

EXERCISE 29.-GREEK-ENGLISH.

1. In difficult matters few companions are faithful. 2. The suppliants touch our knees. 3. Death is a separation of the soul and body. 4. Wealth furnishes men with various aids. 5. Do not yield to the words of wicked men. 6. Do not, my son, be a slave to the service of the body. 7. The Greeks pour cups of milk as libation-offerings to the nymphs. 8. Accustom yourself to, and exercise your body with, toil and sweat. 9. Chatterers vex (or weary) the ear with repetitions (of the same story). 10. Accustom your soul, my son, to good deeds. 11. Evil stories do not lay hold of our ears. 12. We listen with our ears. 13. Do not hate a friend for a small fault. 14. My son, taste the milk. 15. The soldiers bear lances.

EXERCISE 30.-ENGLISH-GREEK.

1. Ω νεανίαι, εθιζετε τα σώματα συν πονῳ και ιδρωτι. 2. Ορεγόμεθα των αγαθών πραγμάτων. 3. Πολλοι τέρπονται χρυσῳ. 4. Εξ αγαθού πράγματος γίγνεται κλέος. 5. Τους καλους μύθους των σοφων θαυμαζομεν, 6. Τα των αγαθών ανθρωπων αγαθα πραγματα θαυμάζεται. 7. Οἱ στρατιῶται μάχονται λόγχαις. 8. Ον διαμειβομαι τον πλούτον της αρετης τους αναξι 9. Mn πείθεσθε τοις λόγοις των φαύλων.

MECHANICS.-XVIII.

LAWS OF FALLING BODIES-PROJECTILES-COLLISION

OR IMPACT.

re two remarkable facts that have been discovered in with the laws of bodies falling down an incline that just notice here. The first is, that if we take any chords, A E, E E, etc. (Fig. 100), all meeting in E, the nt of the circle, and make inclined planes parallel ortional in length to them, a body will take the same fall down each of these inclines. BE, for instance, is longer an DE, yet it is inclined at a much greater

velocity, and it is found that this increase of speed exactly makes up for the greater distance.

The

B

C

The other fact is, that if a body has to fall from one point to another not in the same vertical line, as, for instance, from D to E, the line of quickest descent is not along the straight line joining these two points, but along some curve, as D F E. reason of this is, that if the body be moving down the curve it will at any moment be at a lower level than it would if falling down the incline DE; and since the velocity of a falling body, as we have seen, depends upon the vertical distance passed over, its velocity is all along greater. The space passed over is, however, greater too; but this is more than compensated for by the increased velocity. The curve of shortest descent of all is found to be that which has the greatest curvature without rising as it approaches E. If a pencil be fixed so as to project horizontally from the rim of a wheel, and made to trace a curve on paper while the wheel is rolling on, it will be exactly that of shortest descent. As we shall see further on, there are other remarkable and important properties possessed by this curve, which is called a Cycloid. (See Lessons in Geometry, XXIII., page 309.)

PROJECTILES.

Fig. 100.

Having thus seen the laws which govern the motion of falling bodies, we pass on naturally to notice the movements of proimpedes motion to a greater or less extent. jectiles. Here, of course, as before, the resistance of the air This resistance increases as the square of the velocity, for if the speed of a body be doubled, it not only has to displace twice the bulk of air, but it must do it with twice the velocity, and for this a fourfold force is needed. As, however, our calculations would be much complicated if we took this into consideration, we will neglect it, but we must remember to make allowance for it in our results.

The path of a projectile is in a curve called a parabola, that 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 Lessons in Geometry, XXI., page 251.) We can, however, trace this path in a simpler way.

When a body is projected with any velocity, as, for example, when a bullet is fired from a gun, it is acted upon by two forces -the original velocity with which it was started, which, as we are not considering the resistance of the air, we may consider to be a uniform force; and, secondly, the attraction of the earth, which is an accelerating force, causing it to fall 16 feet in the first second, 48 in the next, and so on. Now from a knowledge of these two motions we can easily tell at what point the body will be at any given moment; and by thus finding several different points in its course we can trace out its path.

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A C and A D represent, then, the two forces acting on the bullet; and, since each produces its full effect, it will at the end of one second have arrived at the point E. Since, however, the force of gravity is not uniform, the line A E, which represents its path, will not be straight, but curved upwards, for when a half of a C has been passed over, gravity will only have caused it to move over a quarter of A D. If now we draw through x a straight line EF, parallel to AB and equal to AC, it will represent the motion of the bullet from its original impulse during the next second. To

represent gravity we must take E G, three times the length of A D, and thus, by completing the parallelogram, we find that at the end of this second the bullet has arrived at H. In the same way, by making H L equal to five times A D, we find K to be the point at which the bullet will have arrived after three seconds, and in this way we can map out its whole path.

Now raise c to the fourth division, and let it fall against D. No momentum will be destroyed; it will merely be shared between the two balls, as much being gained by the one as is lost by the other; and, since both balls have the same weight, each will move with half the velocity that c had on striking D. They will therefore rise together to the first division of the arc D F, for c takes twice as long (E to fall from 4 as it does from 1, and the ve

We see from this the reason why the sights of a rifle are arranged as they are. If the bullet travelled in a perfectly straight line, the soldier would aim directly at the point he wished to hit; but the force of gravity acts on the bullet, and therefore he has to point the rifle at a point as much above it as 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 target, he must aim at a point 64 feet above it. To do this would be very inconvenient and uncertain, as he would be unable to tell whether the point he was aiming at was directly over the mark. The sight at the end next the stock is therefore made to adjust to different elevations above the barrel, according to the distance of the object aimed at; and thus, though the rifleman sees the two sights in a straight line with it, the barrel is really pointed considerably upwards, as will be evident to a bystander.

There is one other fact relating to projectiles, which, though It seems strange, is a necessary result of the second law of motion.

If a body be projected horizontally, no matter how great its velocity be, it will always reach the earth in exactly the same time as if it fell vertically. The speed in falling is not in any way interfered with by the horizontal motion.

COLLISION OR IMPACT.

We said that any force is measured by the velocity generated in a second. There is one class of forces, however, which cannot be so measured, because they do not act for any appreciable length of time. These we call impulses or impulsive forces; any force which is of the nature of a blow is placed in this class.

When one body strikes against another different results will ensue, according to the nature of the bodies. If an ivory ball be allowed to fall on a stone slab, it rebounds or rises from its surface, but the height to which it rises is less than that from which it fell. Were the ball perfectly elastic, it would rise to the same height. This, however, is not all that has occurred, the changes have been more complicated. On striking the slab, the ball is first flattened in a slight degree. In proof of this we may smear the slab with oil, and we shall find the ball marked, not in a minute point as it would be if merely laid on it, but over a space increasing in size with the violence of the blow. The particles are thus compressed, but their own elasticity causes them at once to recover their original position, and in so doing the ball flies up from the slab.

The effect, then, varies with the degree of elasticity of the body. We can, however, only consider the cases of elastic and inelastic bodies, not that any substances are perfectly so, but by examining these we shall get at general principles, which can then be applied or modified as may be required.

We will first consider the case of inelastic bodies, and wellkneaded clay or putty may be chosen as suitable substances to experiment with. Wax, softened with oil, will also answer well. In making experiments on impact, the best plan is to procure balls of the substances chosen, and, having fastened them to strings, suspend them in such a way that they may just touch

one another.

Let us take two such balls, C and D (Fig. 102), of equal weight, and having raised them to the same height, in opposite directions, leave them free to fall together and strike each other. Since both fall from the same height, their velocities are equal, and they each have the same mass; their momenta are therefore equal, and being in opposite directions neutralise each other. Both balls will therefore, after impact, remain at rest.

In order to measure the distance through which the balls fall, we must draw the arcs which they describe and divide them. We do not, however, make the divisions equal, but draw a series of parallel lines at the same distance apart, the lowest being even with the top of the balls, and make our divisions at the points where these cut the arcs. The reason of this is, that the velocity is proportional, not to the length of arc, but to the vertical height, and thus these divisions indicate the velocity.

A

B

F

to the time, therefore
it acquires a double
velocity in falling.
Now whatever velo-
city a body acquires
in falling from any
height, it must start
with that velocity to rise to that height. A velocity, then, half
as great as that acquired by c will raise the two balls to 1.

c

D

Fig. 102.

In the same way, if we make c half as heavy as D, and raising it to the 9th division let it fall, the two will, as before, rise to 1. The mass moved after impact is three times that of c, the velocity will therefore be only one-third as great; they will therefore rise the height. We see thus that when one body } strikes against another, the momentum will be divided between them, and hence the resulting velocity will be as much less than that of the moving body as the mass of the two is greater than its mass.

For example, suppose a ball weighing 1 lb. and moving with a velocity of 60, to impinge against a larger ball weighing 14 lbs. The mass after impact will be 15 lbs., or fifteen times that of the ball; the velocity will therefore be is, or 4 feet per second. No momentum is lost. The original momentum was 1 x 60; after impact it is 15 x 4, which is also equal to 60.

This principle supplies us with a means of measuring very great velocities, as that of a cannon-ball or other missile.

A large block of wood or metal is suspended by a rod so as to swing to and fro with as little friction as possible. This is called a ballistic pendulum. Against this the ball is caused to strike, and by its impact it sets it in motion. A graduated arc is fixed under the block on which the distance to which it swings can be noted, and from this we can calculate the velocity it had immediately after the ball struck it. We have only to measure the vertical height to which it rose, and ascertain the velocity it would attain in falling from that height, and thus we have the velocity with which it started.

The weight of the bullet and the pendulum being also known, we can at once determine the proportion they bear to each other, and thus we can ascertain the velocity of the ball from that of the pendulum.

Suppose, for example, that the pendulum weighs half a ton, and being struck by a ball weighing 24 lbs. is raised to a height of 16 feet. In falling from this height it would acquire a velocity of 32; this, therefore, is that which it had immediately after the ball struck it. But the mass of the ball is to that of the two together as 21 to 1144, or 1 to 48 nearly. The velocity of the ball was therefore 48 × 32, or 1536 feet per second.

Hence we see why, if one body strikes against another, the heavier it is as compared with that against which it strikes, the greater the effect produced. If we want to drive a large nail or to strike a violent blow, we use a heavy hammer, for by it we obtain a much greater momentum, and thus accomplish the work with greater ease. So, too, when we are driving a nail into a plank, we place a support behind or hold a heavy hammer against it. Unless we do this the momentum is shared by the board, which yields to the blow, and thus destroys much of the effect. But when a heavy inelastic body is held behind, this, too, has to share the momentum, and thus the plank yields much less, and the nail is driven more easily.

In the same way some of the feats of strength sometimes exhibited may be explained. A man will lie with his shoulders supported on one chair and his feet on a second. A heavy anvil is then placed on his body, and on this he allows stones to be broken or blows to be struck, which, but for the anvi1 certainly kill him. The reason is, that the moment hammer imparts but a very slight velocity to th

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r. what is the same thing, so that the four quantities shall be proportionals.

It is evident, since a concrete quantity can only be compared with another of the same kind (Obs. 11, Lesson XXVII., Vol. II., of astic page 102), that the fourth quantity determined must be of the in the same kind as the third quantity. In order that the ratios of ough the the two pairs of quantities may be equal, either two must be an peneral laws, of one kind and two of another, or all four must be of the same in together,, kind.

and low it to fall Jhat the momentum aste, they will be ning their shape, being troy the motion of c and refore remain at rest, and

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4 to that which c had. If a
-s suspended so as just to touch one
ad alowed to fall against the
e imparted to the second,
on throughout the entire series,
stroyed by the reaction of the next.
and ball only will rise, all the
So, if two balls be allowed to fall,
We see, then, that no
2 ter ond.
nor than it was in the case of in-
s not shared between all the balls, as it
15 experiments can, of course, be varied
si you are recommended to try them for
ways learnt by seeing or trying a few ex-
Dua by reving out many. As, however, there is
poca ang mai su spon ling ivory balls, the experiments
app way with common glass marbles. Lay
of woodalom; a smooth surface, like the top of a
aad dan so so that a marble may just roll
, or, better still, cut a small groove in which
Ono marble may then be laid in the
la to strike it gently. The latter will
twulo the other will move. The reason
blutely to rest is, that glass is not
y elastic, and thus the reaction is not
putauillent to destroy the motion. If seve-
val marbles be laid so as to touch one another,
anlond ads to strike the end, the same re-
alta will on suo as with the ivory balls.

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Thero is ono other law relating to impact. It, that "the angle of incidence is equal to the ann of reflection." The meaning of this will to olour from the annexed figure. Let any fo A c, in the direction D B, it will reFfrom it ex de diction 1, making the same angle with da me that #D does. The angle DBF, or that elled the angle of in idence, while of 4 fedion, and the law a-sorts that these wp to opties and other branches of wen call fuck bother illustrations of this law.

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יד יד י

LOWEPT TO EXAMPLES IN LESSON XVII.

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40: 37: £95: sum required. Then, by equating the product of the extremes and means, we get the result. We have put the first example, however, in the fractional form, in order to indicate clearly the fact that the ratio of the two quantities of the same kind (acres in this case) is an abstract number, by which the other quantity, the £95, is multiplied. When we state the question in the second way, and talk about multiplying the means and extremes together, some confusion might arise from the idea of multiplying 37 acres by £95. The fact to be borne in mind is that the rule is merely the expression of the fact that the ratios of two pairs of quantities are equal.

5. The example we have given is what is called a case of direct Proportion-that is to say, if one quantity were increased, the corresponding quantity of the other kind would be increased. Thus, if the number of acres were increased, the number of pounds they cost would be increased.

If, however, the case be such that, as one of these corre sponding quantities be increased, the other is proportionally diminished, the case is one of what is called Inverse Proportion. For instance :

EXAMPLE. If 35 men eat a certain quantity of bread in 20 days, how long will it take 50 men to eat it?

Here, evidently, the more men there are, the less time will they take to eat the bread; hence, as the number of men increases, the corresponding quantity of the other kind-viz., the number of days-decreases.

Hence, since 50 men are more than 35 men, the required number of days will be fewer than the 20 days which corre spond to the 35 men.

In stating the proportion, therefore, in order to make the

it than a little over kid foot, and will reach the earth again in ratios equal, if we place the larger of the two terms of one ratio

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6. The last question might also have been solved thus:

35 men eat the quantity in 20 days;

Therefore 1 man eats Therefore 1 man eats

1

of the quantity in 20 days;

Therefore 50 men eat

35 x 20
50
20 x 35

of the quantity in 1 day;

of the quantity in 1 day;

And therefore 50 men would occupy

1

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50 ΟΙ 20 x 35

20 × 35
50

days in

eating the bread. N.B.-To get the time occupied in doing a certain work, when the amount done per unit of time (say per day) is given, we must evidently divide the whole quantity of work by the amount done in a day. In the case given above, the bread being considered the unit, of the bread is eaten in 1 day, and therefore 50 which is the whole amount eaten divided

1

20 × 35

50

20 x 35

GEOMETRICAL PERSPECTIVE.-III. BEFORE proceeding farther and deeper into our subject, we wish to draw the pupil's attention to an explanation of projection, a term applied not only to perspective but also to other systems of representation, namely, orthographic and isometric. Our reason for introducing this now, is in order to make it clearly understood how the plan of an object is to be treated when we are about to make a perspective drawing of that object, as we very frequently meet with cases when the plan of the object to be represented must be drawn according to the position which that object presents, whether horizontal or inclined. The plan, as we said in Lesson I., is produced by perpendicular lines drawn from every part of an object upon a horizontal plane. Now, there can be no difficulty in drawing a plan when the subject represented by it is parallel with the ground or horizontal plane; but it occurs sometimes that it is placed at an angle with both planes, that is, with the picture-plane and ground

by the amount eaten in one day, will be the whole time occupied. plane: therefore in cases of this kind it is necessary to under

7. Hence we get the following statement of Simple or Single Rule of Three.

Write down the ratio of the two quantities which are of the same kind, putting the greater in the first place. Then observing from the nature of the question whether the fourth quantity required will be greater or less than the third one which is given, place the greater of the two in the third place of the proportion, and multiply the extremes and means together.

EXERCISE 51.-EXAMPLES IN SINGLE RULE OF THREE. 1. If 16 barrels of flour cost £28, what will 129 cost?

2. If 641 sheep cost £485 15s., what will 75 cost?

3. If £11 5s. buy 63 pounds of tea, how many can be bought for

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10. If 6 men build a wall in 15 days, how many men would it take
just to finish it in 22 days?

11. If of a ton costs 9s. 8d., what would 42 of a cwt. cost?
12. If a twopenny loaf weighs 1 lb. 2 oz. when wheat is 50s. a
quarter, what should it weigh when wheat sells for 60s. ?
13. If the weight of a cubic inch of distilled water be 255 grains,
and a cubic foot of water weighs 1000 oz. avoirdupois, find the
number of grains in a pound avoirdupois.

14. If 1 lb. avoirdupois weighs 7,000 grains, and 1 lb. troy weighs
5,760 grains, find how many pounds avoirdupois are equal to
175 lbs. troy.

15. Find the rent of 27a. 3r. 15p. at £1 3s. 6d. per acre.
16. The price of standard silver being 5s. 6d. per ounce, how many
shillings are coined out of a pound troy?

17. A bankrupt's assets are £1,500 10s., and he pays 9s. 34d. in the
pound what are his debts?

18. If standard gold is worth 1d. per grain, how many sovereigns
would be coined out of a pound troy of gold ?

19. What is the income of a man who pays 53s. 10d. tax when it
18 7d. in the pound?
30. Raising the income-tax 1d. in the pound increases my amount
of tax by £2 3s. 4d., and the tax I actually pay is £15 3s. 4d.:
what is the rate of the income-tax?
21. A barrel of beer lasts a man and his wife 3 weeks, she drinking
half the amount he does: how long would it last 5 such
couples ?

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stand the first principles of orthographic projection, namely, projection by straight lines upon vertical and horizontal planes. We have mentioned above another method of projection, isometric; as the term has been introduced, we will explain its meaning and then pass it by, as it does not, like orthographic, form any auxiliary to perspective. The term isometric signifies likemeasurement, that is, all the parts of the drawing, both near and distant, are drawn to one and the same scale, also the plan and elevation are combined in one drawing. It is a method much used by architects and engineers when they wish to give what is generally called a bird's-eye view of a building, etc., without diminishing the distant parts, as shown in perspective projection. A drawing made isometrically will enable a strange to understand the proportions, position, and general character of a subject probably better than any other system; hence the reason of its frequent use.

The extent to which we intend to proceed with orthographic projection must be limited to that which relates to, and can assist us in, our present subject, by which we hope to make it a valuable auxiliary in our efforts to render the science of perspective easy and intelligible.

The difference between the results of perspective and orthographic projection is caused by the altered position of the eye only, and from that place is included all that can be seen within when viewing the object. In perspective the eye is in one place the angle of sight. In orthographic projection the eye is supposed to be opposite every part at the same time, above the object when the plan is represented, and before it when the elevation is represented; consequently, in perspective, all the visual rays proceeding from the object to the eye converge to one point; but in orthographic projection these rays are drawn parallel with each other, and perpendicularly to the plane of projection, whether the plane is horizontal or vertical. To make this clear, we request the pupil to compare Figs. 5 and 6 of the last Lesson with Fig. 8, when he will notice that the characteristic difference between the two systems rests entirely upon the different treatment of the lines of projection, which, as we have said, converge in one case, and are parallel in the other. Fig. 8 is to show how a cube is projected orthographically upon vertical and horizontal planes of projection. A is the vertical, and B the horizontal. c is the cube in space, that is, at a distance from angles of the cube perpendicularly to and meeting the plane B, both planes of projection. If straight lines are drawn from the and then lines (a, b, c, d) be drawn to unite them, we shall have a plan of the cube; and as the edges in this case are placed perpendicularly with the ground, the plan will be a square. Again, if horizontal and parallel lines are drawn from the angles of the cube until they meet the vertical plane A, and are then joined by the lines e, f, g, h, we shall produce the elevation; and because the horizontal edges of the cabe are perpendicular to the vertical plane of projection, the drawing in this case also will be a square. Consequently, it will be seen that the drawing of the plan or the elevation is the same size as the object on the respective plane to which the object is parallel, according to the given scale of that object, as in Figs. 10 and 11. This result makes orthographic projection of much importance for practical purposes. The working drawings for the guidance cf builders and mechanists are made by this method. Horizontal lengths and breadths are shown both in the plan and elevation, hut

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