the note (in your mind) into two equal beats or "pulses" of time (you can mark them by beating on the table with your hand); and then the words "Trust in the" will go to the first. beat, and "Lord with" to the second. The large dot above the line shows this division. The words "all thine heart" fall easily to their right notes. To the note Sон you will sing the words" and lean not unto thine." "And" is scarcely heard. Dividing Soн, like the other "reciting note" ME, into two beats (and reciting notes of a chant like this may be divided into as many beats as you please), you will have the words "lean not unto" to the first beat, and "thine alone filling the second. The word "own" you perceive is "slurred" on to two

notes. Take care not to sing the syllables "standing" quickly and sharply. Let them take as much time as the syllables "under" in the same word. The second verse of words, printed underneath, is divided on the same plan as the first. The double bar, you will observe, separates the words of the "reciting note" from those of the "cadence" (as it is called) of the chant. On the upper "staff" you miss the square note for DOH, because the tune begins on ME, and Dон does not occur in the "air." The place of Doн, however, is in the first space, reckoning from the bottom of the staff. On the lower staff it was necessary to make an additional line to carry Soн. This is called a "ledger line."

LESSONS IN PENMANSHIP.-VI.

IN our last lesson we showed our learners how to make the letters m and n by combinations of the top-turn and the topand-bottom-turn. In the present lesson we set before them five Copy-slips for practice, comprising various combinations of the letters m and n with the letters i, u, t, and 1, which they learned to form from the simple bottom-turn. Doubtless many of those who have been endeavouring to teach themselves the art of Penmanship by the aid of our lessons, are now beginning to see that, after all, it is not such a difficult matter to learn to write, since by learning to form only three strokes of a very simple kind, they have found that they have acquired the power of writing six out of the twenty-six letters that form the alphabet. They will soon see that this is by no means the utmost limit of their progress, when in the course of future lessons they discover how many letters there are into whose composition these three elementary strokes also enter.

In preparing paper for copying all our elementary copy-slips in large text, the learner must not omit to insert the diagonal lines as they appear in Copy-slips Nos. 1 to 6 inclusive, as long as he finds them absolutely necessary for the regulation of the slope of his letters, and to enable him to preserve a proper distance between them. To save trouble in ruling sheet after sheet of paper with horizontal and diagonal lines at the proper distances from each other, and to save expense as well, the learner might rule with a sharp-pointed steel instrument, such as a bradawl, one side of a cheap slate with sets of lines similar to those in our copy-slips, taking care, however, to leave a space of one inch and a quarter between each set, to enable him to add the extra horizontal lines that will be required when he begins to make looped letters, and letters such as p and q, that extend below the lower of the two horizontal lines that contain what we have called the body of the letter. A slate thus prepared will be found useful for acquiring facility in forming the various letters, but the learner must by no means omit to write on paper as well.

LESSONS IN ARITHMETIC.-VI. ABRIDGED METHODS OF MULTIPLICATION AND DIVISION. 1. THE methods of multiplication and division explained in the previous lessons are those ordinarily employed; and the learner must make himself perfectly familiar with them before proceeding further.

These processes, however, in particular cases, can often be materially facilitated by various artifices. Some of these shorter methods we subjoin, not only because they are useful in themselves, but because they are valuable as exercises, in explaining the fundamental principles of arithmetic.

2. Any number which is formed by multiplying two or more numbers or factors together is called a composite number.

352

It

has already been explained in a former lesson that the same numbers multiplied together will give the same product, in whatever order the multiplication is effected. Hence, to multiply any number by one which is compositeie., which is composed of several factors-we have only to multiply the number first by one factor, the result by another factor, and so on. Thus, to multiply 352 by 28, since 287 x 4, we can perform the operation as indicated in the margin.

7

2464 4

9856 = 23 × 352

(2.) What will 24 horses cost at 74 crowns apiece. (3.) What will 45 hogsheads of tobacco cost, at 128 crowns a hogshead ?

(4.) What will 54 acres of land cost, at 150 crowns per acre? (5.) At 118 shillings per week, how much will it cost a family to board 49 weeks?

(6.) If a man travel at the rate of 72 miles a day, how far will he travel in 64 days?

(7.) At 163 crowns per ton, how much will 72 tons of lead cost?

(8.) What will 81 pieces of broadcloth cost, at 245 shillings apiece ?

(9.) What will 84 carriages cost, at 384 crowns apiece? (10.) What will a railway 132 miles in length cost, at the rate of £1,960 a mile?

(11.) If I can walk a mile in 16 minutes, how long will it take me to walk 374 miles ?

(2.) How many acres of land, at 35 crowns an acre, can you buy for 4650 crowns.

(3.) A man divided 837 crowns equally among 27 persons, who belonged to three families, each family containing nine persons: how many crowns did each person receive?

(4.) A man bought a quantity of clover seed amounting to 507 pints, which he wished to divide into parcels containing 64 pints each: how many parcels can he make?

5. Multiplying and dividing by powers of 10, and by numbers ending in any number of ciphers.

The products of two tens, three tens, four tens, etc., are called respectively the second, third, fourth, etc., powers of 10. They are 100, 1000, 10000, etc. Thus, the second power is 1 followed by two ciphers, the third 1 followed by three ciphers, and so on; the number of the ciphers in each case being the same as that of the power.

It has been already explained that to multiply by 10, or any power of 10, we have only to annex to the multiplicand the number of ciphers corresponding to the power. Thus, 345 multiplied by 1000 is 345000.

If any number of the right-hand figures in the multiplier be ciphers-as, for instance, in 75000-then, as we have already seen in Lesson IV., Art. 5, we need only multiply the multiplicand by 75, and annex to the product the same number of ciphers, in

this case three.

EXERCISE 12

(1.) Work the following examples in multiplication:

1. 153486 × 10000.
2. 3120467 × 100000.

3. 52690078 × 1000000.
4. 689063157 x 10000000.
5. 4946030506 x 100000000.

6. 87831206507 × 1000000000. 7. 67856005109 x 10000000000. 8. 14376 x 25000. 9. 350634 x 410000. 10. 4630425 × 6200000.

11. 2370000 × 52.
12. 48120000 × 48.
13. 356300000 × 74.
14. 1623000000 × 89.
15. 540000 × 700.

16. 1563800 × 20000.
17. 31230000 x 120000.
18. 5310200 x 3400000.
19. 82065000 x 8100000.
20. 210909000 x 5100000.

(2.) What will 10 boxes of lemons cost, at 63 shillings per box?

(3.) How many bushels of corn will 465 acres of land produce, at 100 bushels per acre?

(4.) Allowing 365 days for a year, how many days are there in 1000 years?

(5.) How much will 50 hogs weigh, at 375 pounds apiece? (6.) If 1 barrel of flour weighs 192 pounds, how much will 500 barrels weigh?

LESSONS IN GEOMETRY.—III. INSTRUMENTS USED IN PRACTICAL GEOMETRY. IN the operations of practical geometry, a case of mathematical instruments must be considered as an essential requisite. These instruments vary in number and quality, according to their price. Some are made of wood, bone, and ivory-as rulers and scales; others are made of brass and steel, German

silver, and other compound metals, such as compasses, drawing pens, and protractors. We shall proceed to describe the most useful, and afterwards to show their application.

The Common Ruler or Straight-edge. This instrument generally consists of the bevelled edge of the plane or diagonal scale, of the common Gunter's scale, of an ordinary foot rule, or of a plain flat rule made with a fine straight edge, for the sole purpose of drawing straight lines from one point to another, or through any two points. It is sometimes made in the form of a right-angled triangle (Fig. 1), with a similar edge, to serve the various purposes of drawing straight lines, perpendiculars, rightangled triangles, and parallel straight lines. In the mechanical arts, a straight line is most readily obFig. 1. tained by fixing a well-chalked string firmly at both ends over the place where it is wanted, on a board or stone, raising it, when tense (i.e., stretched), above the same, and then letting it drop suddenly, when the white or chalky trace of the string will be marked on the board or stone as a straight line.

The Parallel Ruler.-This very useful instrument is constructed in a variety of forms. Those represented in Figs. 2, 3, and 4, are the most common, and the cheapest. The defect of the construction in Fig. 2 is, that in drawing a parallel to a straight line through a given point, if the latter be at a considerable distance from the former, the ruler may, from its

lateral motion, pass the point altogether, and render the problem nugatory. This defect is obviated by the construction in Figs. 3 and 4, provided they be properly managed; but this management is the result of a little practice.

The triangular ruler represented in Fig. 1 being made to slide against a fixed ruler or straight-edge, as represented in Fig. 5, is frequently employed for the purpose of drawing parallel straight lines. In many cases this apparatus will be found even more handy for this purpose than the parallel rulers represented above. Fig. 5 represents the same triangle in two different positions, and not two separate triangles.

In order to test the accuracy of a ruler, let it be applied to one eye, and viewed along its edge from one end to the other; the slightest departure from the straight line will then become visible. A good ruler, besides having a straight edge, must be perfectly flat and even, flexible, and made of wellseasoned wood. Some are made of ivory, bone, and metal; these are less liable to be affected by changes in temperature, or by the humidity of the atmosphere. Parallel straight lines are most easily drawn by artists and mechanics, with an F or a T square, of which the form is distinctly noted by the name.

Fig. 5.

The Compasses.-Of compasses there are several kinds. This instrument, which usually consists of two equal legs jointed at one extremity, is employed for measuring the lengths of straight lines, measuring and laying off distances, and describing circles or arcs of circles in general. The Dividers, or compasses with dry points, represented in Fig. 6, are chiefly used for dividing straight lines into equal parts, or into parts having any other proportion to each other. The best kind are furnished with a turnscrew for tightening the screw-axle at the joint. Others

Fig. 7.

pencil, or in mere trace. The tracingpoint in Fig. 7 is furnished with a joint and a screw, in order to keep it perpenFig. 6. dicular to the paper when the legs are stretched to a great length. The ink-point, represented in Fig. 8, is furnished with a screw, to admit more or less ink at pleasure, with a joint for the same purpose as the tracer, and with a joint in one of the leaves of the point to admit of its being cleaned. The pencil or crayon-point, represented in Fig. 9, is furnished with a joint for keeping the pencil or crayon perpendicular to the paper, and a socket or case for holding it. The socket compasses are also furnished with a lengthening bar, represented in Fig. 10, which is furnished with a socket exactly the same as that of the leg, in order to admit of the description, that is, the drawing of larger circles than those which can be drawn only by the use of the movable points and legs of the compasses.

The Bow Compasses, so called because in their first construction they could be shut up into a hoop, which served as a handle to them; or the Plug Compasses, represented in Fig. 11, and so called because the stationary leg screws out and in like a plug, are only used for describing circles of a very small size. Such compasses are of the greatest utility to draughts- Fig. 11. men and engineers in drawing their plans. The plug construction seems to present some advantages over the old bows.

Fig. 12.

Beam Compasses are employed for describing circles of very large radius, and such as are far beyond the reach of a case of mathematical instruments. They consist of a long beam or bar, carrying two brass cursors, that is, pieces on which it runs. One of these is fixed at one end, and the other slides along the beam, and is furnished with a screw to fix it at any required distance. To the cursors may be screwed points of any kind, whether steel tracers, pencils, or crayons, or ink points. This apparatus is represented in Fig. 12. To the fixed cursor there is sometimes applied an adjusting or micrometer screw, as seen in the figure, to enable a given distance or radius to be taken with the greatest nicety.

In a case of mathematical instruments are also contained a Tracer and Drawing Pen, for drawing straight lines in trace, or in ink. These two are usually joined in one instrument, the tracing point being screwed into the drawing pen; this instrument is represented in Fig. 13, where the ink-point is constructed exactly on the same principle as that of the socket compasses. In choosing a drawing pen, it is better to select one which has an ink-point made of German silver. The steel ink-points are apt to get rusty if they are not kept carefully wiped, and lines Fig. 13. drawn in red ink with a steel-pointed drawing pen soon get discoloured, owing to the action of the ink on the metal while in the pen.