LESSONS IN PENMANSHIP.-IX.

THE letter p is the first letter that the learner has met with that extends below the line b b, and it will be necessary here to say something about its proportions, as they are given in Copyslips Nos. 28, 30, and 31.

It will be remembered that in "large text," the distance between the lines a a, b b, that contain what we have called the body of the letter, is, or ought to be, exactly half an inch; and as the line c c is midway between the lines a a, b b, the distance

f

On inspecting any copy-slip that has the letter 1 in it, it will be found that the letter p extends as far below the line b b as the letter 1 extends above the line a a. That portion of the letter p which extends above the line a a is longer by one-sixteenth of an inch than the distance to which the letter t extends above the same line, or the distance between the top of the bottom-turn of the letter i and the dot above it, as may be seen by examining Copy-slips Nos. 30 and 31.

We have been thus particular in dwelling upon the distances to which letters such as t, 1, h, p should extend above a a, or

between each of these lines and the central line, c c, is a quarter of an inch. Now the distance between a a and the line ff, at which the long straight stroke of the letter p is commenced, is also a quarter of an inch, and is equal to the distance between a a and c c, or between b b and c c. The distance between bb and g g, the line at which the long straight stroke terminates, is rather less than half an inch, or, to speak in exact terms, just seven-sixteenths of an inch; that is to say, if an inch were divided into sixteen equal parts, the distance between bb and g g is equal to seven of them, while the distances ƒ a, a c, c b, on the straight line fg, are each equal to four-sixteenths of an inch, which is merely another expression for a quarter of an inch, as our learners will find when they have got on far enough in Arithmetic to be working at Vulgar Fractions.

below bb, in order to induce the learner to pay strict attention to the relative proportions of his letters. The importance of this will be seen by any one who is curious enough to extend these letters to a greater or less length above a a, or below b b, than is allotted for their extension in our Copy-slips. The general appearance of handwriting that would otherwise be good, is often completely spoiled by a want of proper proportion in the heads, loops, and tails of the letters. Those who wish to be distinguished for writing a plain and legible hand, must aim at the neatness and beauty of the writing that is found in old deeds, and books copied by the monks who lived before the time of Caxton. The letters of these famous penmen are as regular in their proportions and as sharply and delicately defined as if they had been carefully printed from well-cut type.

LESSONS IN ARITHMETIC.-IX.

LEAST COMMON MULTIPLE.

1. ONE number is called a multiple of another when it can be divided by the latter without a remainder.

Thus, a measure and a multiple are the converse of each other. If a number divides another without remainder, it is said to be a measure of it, and the latter number is said to be a multiple of the first.

A common multiple of two or more numbers is a number which can be divided by cach of them without a remainder. It will clearly be a composite number, of which each of the given numbers must be a factor, for it could not otherwise be divided by them.

The same numbers may clearly have an infinite number of common multiples, for any one common multiple having been found, another may be obtained by multiplying it by any number.

The continued product of two or more numbers will always give a common multiple of those numbers.

The least common multiple of two or more numbers is the least number which can be divided by each of them without a remainder.

Thus, 70 is the least common multiple of 2, 5, and 35.

2. The least common multiple of two or more numbers is evidently composed of the continued product of all the different prime factors which compose the given numbers, each one being repeated as often as the greatest number of times it occurs in any one of the numbers. For if it did not contain all the prime factors of any one of the numbers, it could not be divided by that number.

On the other hand, if any prime factor were employed more times than it is repeated in any one of the given numbers, it would not be the least common multiple.

For the sake of brevity the words "least common multiple" are sometimes written L. C. M.

3. EXAMPLE. Find the L. C. M. of 12, 126, and 735. These are respectively equal to

2 x 2 x 3, 2 x 3 x 3 x 7, 3 x 5 x 7 x 7. Now 2, 3, 5, 7 are all the different prime factors which occur in any of the numbers; and the greatest number of times which 2 occurs is twice-namely, in the first; the greatest number which 3 occurs is twice-namely, in the second; 5 only occurs once-namely, in the third; and the greatest number of times which 7 occurs is twice-namely, in the third. Hence the L. C. M. required will be

2 x 2 x 3 x 3 x 5 x 7 x 7; that is, 8820.

4. The process, then, of finding the least common multiple of two or more numbers is reduced to that of splitting up the numbers into their prime factors.

This may be effected, however, by a more convenient method of arrangement than splitting each number separately into factors would be, for which we give the following

numbers.

Rule for finding the least common multiple of two or more Write down the numbers in a straight line apart from each other. Divide by the least number which is a measure of two or more of them, and set down the quotients and the undivided numbers in a line below. Take again the least number which is a measure of two or more of these numbers last set down, and perform the same operation as before. Continue it until there are no two numbers which can be divided by any number greater than unity. The continued product of all the divisors, and the numbers set down in the last line, will be the least common multiple required.

5. EXAMPLE.-To find the L. C. M. of 12, 42, 72, and 84.

The process will be sufficiently understood from the following working::

2) 12, 42, 72, 84

2) 6, 21, 36, 42

3) 3, 21, 18, 21

7) 1, 7, 6, 7

1, 1, 6, 1

Hence the L. C. M. is 6 x 7 x 3 x 2 x 2; that is 504.

This method of arrangement evidently gives the greatest number of times which each prime factor occurs in any one of the given numbers. Thus 2 occurs three times in 72, 3 occurs twice in 72, and 7 occurs only once-viz., in 42 and 84.

EXERCISE 21.

1. Find the least common multiple of the following numbers:1. 15 and 45. 13. 1, 2, 3, 4, 5, 6, 7, 8, and 9. 14. 657, 350, 876, 1095, 2190, and 5795.

2. 63 and 18.

3. 6, 9, and 15.

4. 8, 16, 18, and 24.

5. 9, 15, 12, 6, and 5. 6. 5, 10, 8, 18, and 15. 7. 24, 16, 18, and 20. 8. 36, 25, 60, 72, and 35. 9. 27, 54, 81, 14, and 63. 10. 72, 120, 180, 24, and 36. 11. 375, 850, 3400, and 5085. 12. 7, 11, 13, and 5.

15. 42, 12, 84, and 72, 16. 9, 12, 72, 36, and 144. 17. 8, 12, 20, 24, and 25. 18. 63, 12, 84, and 7. 19. 54, 81, 63, and 14. 20. 75, 120, and 300. 21. 96, 144, and 720. 22. 256, 512, and 1728, 23. 375, 850, and 3400.

LESSONS IN GERMAN.-VIII. SECTION XVI.-USE OF THE DEFINITE ARTICLE; PROPER NAMES, ETC ETC.

THE plural of Mann is Männer; except in compounds, where it is generally Leute ($ XV. Note), as Lantmann, countryman: Landfeute, country-people. Zimmermann carpenter; 3immerleute, carpenters. Hauptmann, captain; Hauptleute, captains. Kaufmann, merchant; Kaufleute, merchants.

Bolf corresponds mainly to our word people. Unlike this, however, it has different forms for the two numbers, as :-Die Franzojen sint ein lebhaftes Volk; the French are a lively people. Die Fürsten schwelgen, und das Volk leitet ; the princes revel, and the people suffer. Alle Völker auf Erden, 1 Moses xviii. 18; all the nations of the earth, Genesis xviii. 18.

The word one, as a pronoun, is, in English, often inserted after an adjective, to avoid the repetition of the noun; in German, however, the adjective in such a case stands alone, as:Er hat einen guten Hut, und ich habe einen schlechten; he has a good hat, and I have a bad (one). Ich habe gute Hüte, und er hat schlechte; have good hats, and he has bad (ones). Er hat guten Wein, und ich habe schlechten, he has good wine, and I have bad (wine).

The adjective and participle preceded by an article are often used substantively, as well in the singular as in the plural, as:Der Zufriedene (Sect. IX. 2) ift glücklich; the contented (man) is happy. Die Zufriedene ist glüdlich; the contented (woman) is happy. Die Zufriedenen find glücklich; the contented are happy. Ein Zufriedener (Sect. X.) ist glücklich; a contented (man) is happy. Der Sterbente, die Sterbente; the dying (man), the dying (woman). Die Lebenten: the living.

article, converted into abstract nouns, as :-( 1. Adjectives in German are often, by means of the definite -Er verehrt das Schöne: he adores the beautiful.

2. The use of the definite article before nouns, taken in a generai sense, is much more frequent than in English, as:-Der Tiger ft flint: the tiger is agile. Der Diamant ist ein Erelstein; the diamond is a precious stone. Das Gold ist ein ebels Metall; (the) gold is a precious metal. Die Luft ist ein Element; the air is an Die Seele ift unsterblich; the soul is immortal. Das Wasser ist ein Element: (the) water is an element. Der Mensch ist fterblich; (the) man is mortal. Die Faulheit ist ein Lafter; (the) idle

element.

ness is a vice.

The plural is used in the same manner, as :(the) tigers are agile.

-Die Tiger find flink;

3. The definite article is sometimes used instead of the pos sessive pronouns, as-Er hat ein Buch in der Hant; he has a with the (its) father. book in the (his) hand. Das Kind ist bei dem Bater; the child is

Der

4. Proper names and titles are often preceded by the defi nite article, as :-Wo ist ter Heinrich? where is (the) Henry? Kaiser Heinrich, the Emperor Henry. Der König Heinrich; (the) King Henry.

The definite article likewise commonly precedes the adjective qualifying a proper name, as :-Die schöne Helene, the beautiful Helen. Der arme Richard; (the) poor Richard.

The article is also generally used before the word Schule, Kirche, Markt, Müble. 2c., as:-Er ist in der Schule; he is (in) at (the) school. Er ist in der Kirche, he is (in) at (the) church. Grift auf

longer whilst we make a few remarks upon some of the peculiarities of Angular Perspective. No doubt it is much more difficult to understand than parallel perspective, arising from the great variety of positions in which objects may be placed, for if the lines are, on the one hand, ever so slightly out of the perpendicular from the picture plane, or, on the other, in the least degree vary from the parallel to the picture plane, the treatment necessarily comes under the rules of Angular Per

ing over your work in an ungraceful or painful attitude. The ye should be as nearly as possible directly opposite the centre of your drawing, and the inclination of your paper and copy should be such that a line passing from your eye to either paper or copy, when you are at work, should pass through the centre of the copy at a in Fig. 35, or the centre of the drawing at b, as nearly as possible at right angles to their respective planes. It is unnecessary to give directions as to the manner of holding your pencil. Your own judgment must direct you in a great measure as to that. It matters little, so that you feel the instrument fit your fingers easily. If proper attention has been bestowed upon the primary instructions that we have given, you have already learned the importance of depending not solely on your fingers, but also on the action of the wrist and arm. The hand should not be suffered to rest upon the paper upon which you are drawing, if it can be avoided, but have a spare piece to lay under it while you are at work. It will serve another purpose-to try the points of your pencils upon, or the points of your pens, crayons,

and brushes when you are sufficiently advanced to draw with, pen and ink, or to paint in water-colours. Begin at once to preserve your drawings in a portfolio. Even when you have failed n many attempts you should keep them by you. Destroy nothing that you do, and you will soon learn to do nothing that you would desire to destroy. Preserve order in the disposition of all your materials: much time and vexation may be saved by Fig. 40. Fig. 41.

spective. Let us try to make this clear by the help of Figs. 36 and 37.

Fig. 36 is a case of parallel perspective (see p. 72, Fig. 27a). Fig. 37 has its sides a b and ef slightly removed from the perpendicular c d, and the sides a e and bf in the same proportion removed from the line e g, parallel to PP; consequently it presents the angle at e to the picture plane. (We shall presently be under the necessity of seeking a little help from Geometrical Perspective, in order to make ourselves more clearly understood.) There is another peculiarity in connection with this relating to the position of the vanishing points; we request

the pupil to look at Fig. 38: he will observe that the angle of the building, a b, is nearest the eye, whilst the side a b c d retires one way to v P 1, and the side a bef retires another way to v P 2. Now, when he sees this he will probably say, "Yes, these sides certainly do retire as so stated, but I should like to be informed why these two vanishing points are placed where they are. Is there any rule for so placing them? or is it

it; and, above all things, remember that what is worth doing, is worth doing well.

We propose now to give some instructions in Angular Perpective; we use this term when the object presents an angle, and not a side, to the picture plane, that is, when the angle is nearest us and all the sides retire; this occurs especially when all rectangular forms, such as buildings, boxes, and things of a similar shape and character, are so arranged. (See p. 72, Fig. 276.) Before we proceed to explain the method of drawing objects so placed before us, we must detain the pupil a little

merely a matter of choice?-in short, can I place them anywhere I please ?" These are very fair questions, and we will en deavour to answer them. Of course, the house (Fig. 38) must have a ground-plan, which will be placed with regard to the picture plane as it is shown in Fig. 39, the angle towards us, and the sides retiring. Now let us suppose we are standing at s P (station point), from which place we are to make our drawing; from this place we determine our vanishing points, and the distance these vanishing points are apart will determine whether we are near or at a greater distance from the object. Then to

determine our vanishing points, we must give the following geometrical rule :-"Draw a line from the station point, parallel to the ground plan as far as the plane of the picture, from which draw a perpendicular line to the horizontal line (line of sight); this will give the vanishing point." Let us look at Fig. 39, we shall find that the line a b is drawn from s P 1 to the picture plane, parallel to one of the retiring lines of the ground plan, cd, which gives v P 1; also a e is drawn parallel to fc, the other retiring line of the ground plan which gives us v P 2. But if the station point had been further off, as at S P 2, the line ab would have been from 8 P 2 to h; therefore at h would then be found v P 1; so on the other side the VP 2 would have been at i. Suppose the station point were placed at SP 3, then the vanishing points would be nearer each other. So it will be seen, the further the vanishing points are apart, the further we are from the object; and the nearer we are to the object the nearer together are the vanishing points. Our object,

points: for instance, let him trace out the lines a e and b fin Fig. 38, they will meet at v P 2; and the lines on the other side, a c and b d, will meet at v P 1. In an engraving, the vanishing points for all horizontal retiring lines may be found in this way, and they will determine also the line of sight which runs through these points. If he discover that these horizontal retiring lines do not meet in the same point, it will be because they are not parallel retiring lines; that is, the objects themselves are not placed in a parallel position with each other. This leads to another observation connected with this last remark; if there are fifty retiring lines, and all parallel, there will be only one vanishing point for them all; but if amongst these fifty there are not two parallel, there will be fifty vanishing points. As the pupil, we hope, will clearly comprehend this interesting feature in perspective drawing, he may apply the rule when he has an engraving before him. We know that the rectangular tops and bottoms of windows and doors are horizontal, and

then, in giving this little explanation, is to account for placing the vanishing points. To carry this rule out by producing a drawing of the elevation of the house from the ground plan, will be considered hereafter.

By this explanation we only undertake to satisfy our pupils that we can make a correct drawing of the building if the vanishing point was at h, b, or k; only observe, that if we approached too near, the angle of sight, m, would be too large, so much so as probably to become as great, or greater than 60°. (See p. 72.) While writing these remarks on Angular Perspective, we found that it was absolutely necessary to give these geometrical reasons for the positions of the vanishing points, because as many lines in a picture retire and vanish elsewhere than at the point of sight, we felt bound to give these reasons, which need not cause the pupil to imagine there is anything to discourage him, as the mode of finding them in a picture, as well as when drawing from Nature, is very simple. Here, then, the pupil may ask, "If I have a drawing before me to copy, in which the vanishing points are not marked, how shall I find them?" Let us suppose the copy is an engraving (and the vanishing points are never shown in engravings), let him trace out the retiring lines in the picture-we think there can be no difficulty in recognising them-these lines traced out will give the vanishing

parallel with the eaves and horizontal ridges of roofs, the courses of the bricks, etc. Let him trace out as many of these lines as he can, if he understands they are intended, as in the object itself, to go off in the same direction, and he will find them meet at the same vanishing point, and soon discover whether the engraving is correct or not in the grammar. The uneducated eye may not detect small faults in the general appearance of the engraving, and thousands of drawings and paintings by really clever artists pass muster, and are admired, although they may be full of mistakes; just as in speaking, the grammatical errors habitually made by uneducated men are not even known to be such among themselves, but an educated man will notice them, although he may not remark upon them. After the pupil has discovered the vanishing point for the horizontal retiring lines in the engraving, he will then have found the position of the line of sight; then, in making his drawing, he must begin by placing this V P on his paper, and proceed by marking in the nearest line to the vanishing point, and so on, line after line, as we have before said. We know from experience the great advantage of this method, and have frequently remarked the rapid progress that has been made by those who have feared that drawing was an art too difficult for them to attain.

The method of drawing Fig. 38 will be as follows:-Draw the