represent gravity we must take EG, 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.

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. 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 bo 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 ver 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 clastic 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

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 velocity is proportional to the time, therefore it acquires a double velocity in falling. Now whatever velocity 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.

Fig. 102.

D

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 1, or 4 feet per second. No momentum is lost. The original momentum was 1 x 60; after impact it is 15 × 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 24 to 1144, or 1 to 48 nearly. The velocity of the ball was therefore 48 x 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 rail 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 anvil, must certainly kill him. The reason is, that the momentum of the hammer imparts but a very slight velocity to the anvil, on

account of the greatly superior weight of the latter. This small velocity is easily overcome by the muscles, which being stretched, act, to a certain extent, like a spring, and thus the blow is scarcely felt.

We must now pass on to consider the impact of elastic bodies, and for this we may take balls of ivory suspended in the same way as those of clay were. We shall find that, though the effects produced by these are different, the same general laws apply. The bodies, however, instead of moving on together, will, after impact, rebound and fly apart.

Let us raise one of the balls c (Fig. 102) and allow it to fall against the other. The first effect will be that the momentum will be shared between the two, but, being elastic, they will be compressed, and the reaction in regaining their shape, being equal and opposite to the action, will destroy the motion of c and double that of D. The former will therefore remain at rest, and D will move on with a velocity equal to that which c had. If a series of several balls be thus suspended so as just to touch one another, and the end one raised and allowed to fall against the others, the motion of the first will be imparted to the second, and by that to the third, and so on throughout the entire series, the motion of each being destroyed by the reaction of the next. The result will thus be that the end ball only will rise, all the others remaining at rest. So, if two balls bo allowed to fall, two will be raised at the other end. We see, then, that no momentum is lost here, any more than it was in the case of inelastic bodies; but it is not shared between all the balls, as it was in the other case. These experiments can, of course, be varied to almost any extent, and you are recommended to try them for yourselves, for more is always learnt by seeing or trying a few experiments than by reading about many. As, however, there is difficulty in procuring and suspending ivory balls, the experiments can be tried in a simpler way with common glass marbles. Lay two thin strips of wood along a smooth surface, like the top of a table, and adjust their distance so that a marble may just roll along between them; or, better still, cut a small groove in which the marbles may run. One marble may then be laid in the groove, and another made to strike it gently. The latter will come almost to rest, while the other will move. The reason why it does not come absolutely to rest is, that glass is not perfectly elastic, and thus the reaction is not quite sufficient to destroy the motion. If several marbles be laid so as to touch one another, E and one made to strike the end, the same results will ensue as with the ivory balls.

K

There is one other law relating to impact. It is, that "the angle of incidence is equal to the Fig. 103. angle of reflection." The meaning of this will be clear from the annexed figure. Let any body strike against a surface A c, in the direction D B, it will rebound from it in the direction B E, making the same angle with the perpendicular BF that BD does. The angle DBF, or that at which it strikes A C, is called the angle of incidence, while FBE is the angle of reflection, and the law asserts that these are always equal. As we pass to optics and other branches of physics, we shall find further illustrations of this law.

ANSWERS TO EXAMPLES IN LESSON XVII.

1. It will rise a little over 156 feet, and will reach the earth again in 6 seconds.

2. The elevation is 53 × 16, which equals 30 x16, or 484 feet.

3. It will strike the earth with a velocity of 16).

4. It will take 7 seconds, in the last of which it will fall 208 feet. 5. 16 x 33, or 528 feet.

6. It would require 6 seconds, and pass over 576 feet.

LESSONS IN ARITHMETIC.-XXXII.

RULE OF THREE-SINGLE AND DOUBLE.

1. THIS is a name given to the application of the principles of Simple Proportion to concrete quantities. We have shown (Art. 5, Lesson XX., Vol. I., page 343) that if any three numbers be given, a fourth can always be found such that the four numbers shall be proportionals. Hence, if three concreta quantities be given, two of which are of the same kind, and the third of another kind, a fourth quantity of the same kind as the third can be found such that it shall bear the same ratio to the third quantity as the first two bear to each other;

or, 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, page 102), that the fourth quantity determined must be of the same kind as the third quantity. In order that the ratios of the two pairs of quantities may be equal, either two must be of one kind and two of another, or all four must be of the same kind.

2. Suppose we have the following question proposed :--EXAMPLE. If the rent of 40 acres of land be £95, what will be the rent of 37 acres ?

It is evident that the sum required must bear the same ratio to £95 that 37 acres do to 40 acres.

Hence we have, writing the ratios in the form of fractions, Sum required. 37 acres the abstract number £95 40 acres Therefore the sum required 2x £95, which can be reduced to pounds, shillings, and pence.

=

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 ccnfusion 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 correspond to the 35 men.

In stating the proportion, therefore, in order to make the 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 the other ratio in the third place.

Thus, placing 50 men in the first place, we must put 20 days (which, we can see, will be larger than the required answer) in the third place, and then the statement would be correctly made thus:

50 : 35 :: 20 days 00 required number of days. Therefore the required number of days 20 × 5 days = 14 days. N.B. We might reduce the example to a case of Direct Proportion thus, which will, perhaps, explain the above method more clearly:of the bread in one day.

35 mon eat 50 "9

required number of days the number of men, we haveHence, since the quantity eaten in one day will increase with

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 overy 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

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

£385 ?

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 GOs. ? 13. If the weight of a cubic inch of distilled water be 25373 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 5,700 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 £1,500 108., and he pays 9s. 31d. 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 is 7d. in the pound? 20. 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 ?

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 stranger 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 per spective easy and intelligible.

The difference between the results of perspective and ortho. graphic projection is caused by the altered position of the eye only, and from that place is included all that can be scen 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 в 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 aro then joined by the lines e, f, g, h, we shall produce the elevation; and because the horizontal edges of the cube 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 of builders and mechanists are made by this method. Horizontal lengths and breadths are shown both in the plan and elevation, but

heights are indicated only in the elevations. Sometimes when the subject is a simple one-for instance, a plain wall-its course and thickness will be shown in the plan, and its height marked by indices in brackets at the end, as (10.5 feet), meaning that it is to be built 10 feet 6 inches high. Fig. 9 is the plan and front elevation of a cottage. It will be seen that if the plan be drawn first, perpendicularly dotted lines must be drawn parallel with each other from every angle, and from the terminations and projections of each line, which will determine the extent of the elevation and of its several parts, but not its height. If the elevation be drawn first, the perpendicularly dotted lines are projected downwards to produce the plan. In orthographic projection we usually draw a line to represent the meeting or axis of the two planes of projection, the horizonal and the vertical, which, as in Fig. 10, we have marked ry; therefore it must be remembered that all above that line is understood to be the vertical plane of projection upon which the elevations are drawn, and all below it the horizontal plane upon which the plans are drawn. The plan of a circle when parallel with the ground is a circle of the same size indicated by the scale. The elevation is a straight line only, equal to the diameter (Fig. 10). If the circle is standing on its edge perpendicularly to the ground, then its plan is a straight line only, and the elevation is a circle (Fig. 11). To illustrate the positions (Fig. 10), let the pupil hold a pennypiece horizontally before, and level with, his eyes; he will see the edge, the elevation; then let him place it upon the ground, and look down upon it; he will see the whole circumference, the plan. Reverse the position of the penny, and do the same for Fig. 11. We trust there will be no difficulty now in understanding the position of the eye with respect to both planes of projection. As we intend to devote the present Lesson to the consideration of this subject, preparatory to more important questions in perspective, we will give our pupils a few simple problems for practice, reserving others of a more complicated nature till they are required in future Lessons.

PROBLEM II. (Fig. 12).—A rod, 4 feet long, is parallel with, and 2 feet from, both planes; draw its plan and elevation. Scale inch to the foot.-First draw ay, the axis of the planes, and draw ab, 4 feet long, parallel with and 2 feet from xy; then from the extremities a and b draw perpendicular lines to c and d; mark c and d 2 feet above a y, and join them; e will be the elevation, and ƒ the plan.

PROBLEM III. (Fig. 13).—When the same rod is at an angle of 40° with the vertical plane and parallel with the horizontal plane.-Draw a line eg at an angle of 40° with xy, make ef equal to 2 feet, and draw fa parallel to a y: a will be the plan of one end of the rod 2 feet from the vertical plane; upon eg and from a make a b, the plan, equal to 4 feet: draw the perpendicular lines a c and bd, and draw cd, the elevation, parallel with and 2 feet above xy.

PROBLEM IV. (Fig. 14).—When a rod is at an angle of 40° with the ground and parallel with the vertical plane.-Draw eg at an angle of 40° with ey, and draw the perpendicular ef 2 feet from xy, also fc parallel with xy; cut off cd, equal to 4 feet, the whole extent of the rod: from c and d draw perpendiculars cutting ay to a and b; join ab, for the plan, parallel with xy.

When the object is at an angle with both planes, the angle of

inclination with the horizon is made on the horizontal plane.

PROBLEM V. (Fig. 15).-Let the rod have one end on the ground, and let it rise at an inclination of 50°, and let its plan be at an angle of 40° with the vertical plane.-Draw the line cag at the given angle 40° with the vertical plane; upon this line the plan will be represented. Draw a h at an angle of 50° with a g, and make a m equal to the length of the rod; from m draw mn perpendicular to ag; an will then be the plan of the rod when inclined to the horizon at 50°. Draw ned and ab at right angles with a y, and make cd equal to mn; join bd; the line bd will be the vertical elevation. That this may be more clearly understood, we will draw the eidograph of the problem, Fig. 16, that is, the figure or appearance it would present when placed in conjunction with the two planes of projection (Fig. 8 is also an eidograph). In Fig. 16 ao is the given rod, and an is its plan. Now in order to get the inclination of a o, the rod, which is raised from the paper at an inclination of 50°, must be rabatted, that is, thrown down upon the horizontal plane; the course of the dotted arc o m will show this. We must construct the angle of the inclination of the rod upon the horizontal plane, that is, the

angle it forms with the ground; therefore man will be equal to o an; this was the reason the angle man in Fig. 15 was made 50°. By comparing Figs. 15 and 16, the same letters being used in both, the corresponding lines will be seen, and it will be understood why c d in Fig. 15 is made equal to mn, because, as in Fig. 16, mn is equal to no, the distance of the upper end of the rod from the ground, and no is equal to cd, therefore m n is equal to c d.

PROBLEM VI. (Fig. 17).-The frustrum of a right square pyramid rests with its base on a horizontal plane, the lengths of the edges of the top and base being respectively 13 and 24 inches, and the height 2.8 inches; draw its plan and elevation.—If a pyramid be divided into two parts by a plane parallel to its base, the part next the base is called a frustrum of a pyramid, or sometimes a truncated pyramid. Draw the square a b d c, the plan of the base 2:4 inches side (ɛee Lessons in Geometry, Problem XVIII., Vol. I., page 255), and within it the square efhg, the plan of the top 1.3 inch side. In order to place the plan of the top so that the edges shall be equidistant from the edges of the plan of the base, proceed as follows:-Draw the diagonals cb and ad, make cn equal to 1.3 inch, and draw nh parallel to cgfb; draw gh parallel to cd; the rest will be evident, as the angles are in the diagonals, and the sides are parallel to a b and a c respectively. Having drawn the plans, then draw xy, the ground line, parallel to one side of the square; draw am and bl; draw the lines ei and ƒ k, continuing them above a y equal to the height of the frustrum 2.8 inches; join im, kl, and i k; mikl will be the elevation. The pupil will observe that other elevations can be drawn from the same plan, opposite any other side, when required for working purposes-a common practice in drawing extra elevations for building construction; in these cases all that is necessary is to arrange the ground line or axis of the planes opposite the side of which the elevation is required. Fig. 18 is the same subject as Fig. 17: xy is placed parallel to one of the diagonals of the plan, consequently two faces of the frustrum are seen, a' and b', shown in the plan as a and b.

imperative, aic, ayons, ayez; être, to be, sois, soyons, soyez 5 3. Exceptions.-Avoir, to have, make in those persons of the savoir, to know, sache, sachons, sachez; and aller, va, and vas before ม not followed by an infinitive.

4. Vouloir has only the second person plural, veuillez, have the goodness to....

tive by most of the French grammarians. These parts, however, 5. A third person singular and plural is given in the imperabelong properly to the subjunctive, as they express rather a strong wish than a command. The English expressions, let him speak, that he may speak, are rendered in French by qu'il parle. English to the right, to the left. 6. A droite, à gauche, correspond in signification to the Allez à droite, à gauche,

Go to the right, to the left.

7. For the place of the pronouns in connection with the imperative, see Sect. XXVI., 1, 4; Sect. XXVII. 1, 2, 3, 4. RÉSUMÉ OF EXAMPLES. Prenons la première rue à droite. Ne cherchez plus à le tromper. Sachons nous contenter du néces. saire.

Let us take the first street to the right.
Seck no longer to deceive him.
Let us know how to content ourselves
with necessaries.