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Parallel straight lines are such as lie in the same plane. and which though produced ever so far both ways, Fig. 3. do not meet (fig. 3). An angle is the inclination of two straight lines to each other, which meet in a point, and are not in the same direction. The pointin which they meet is called their vertez, and each of them is called a side or leg of the angle. The angle itself is generally called a plane rectilineal angle, because it necessarily lies in a plane, and is formed of straight lines. Curwilineal angles are such as are formed on the surface of a sphere or globe; but the consideration of such angles belongs to the higher geometry. The magnitudes of angles do not depend on the lengths of their legs or sides, but on the degree or amount of aperture between them, taken at the same distance from the vertex. An angle is generally represented by three letters, one of which is always placed at the verter, to distinguish it particularly from every other angle in a given figure, and the other two Fig. 4. are placed somewhere on the legs of the angle, * but generally at their extremities; and in reading or in speaking of the angle, the A letter at the vertex is always placed between so the other two, and uttered or written accordingly. Thus, in fig. 4, which represents an angle, the name of the angle is either B.A. c or c A B: the point A is called its vertex; and the straight lines B.A, C A, its sides or legs. Angles are divided into two kinds, right and oblique, and oblique angles are divided into two species, acute and obtuse. When one straight line meets another, at any point between its extremities, and makes the adjacent or contiguous angles equal to each other, each of them is called a right angle, and the legs of each of these angles are said to be Fig. 5. perpendicular to one another. Thus, in fig. 5, the straightline A Bomeets the straight B line c D in the point A, and makes the adjacent angles c A B, D A B, equal to each other; each of these angles is therefore called a right angle; and the straightline A B is said to be perpendicular to the straight line A c, or DA, and consequently A C or A D is perpendicular to A B. en one straight line meets another, at any point between its extremities, and makes the adjacent angles unequal to each other, each of them is called an oblique o: ; that which is Fig. 6. greater than a rightangle is called B an obtuse angle : and that which is less than a right angle is called an acute angle. Thus, in fig. 6, the straightline An meets the straight line op, in the point A, and makes ld . A o the adjacent angles unequal to each other; each of these angles is therefore called an obtuse angle ; the angle c A R, which is greater than a right angle, is called obtuse: and the angle D A B, which is less than a right angle, is called acute. A plane figure in geometry, is a portion of a plane surface, inclosed by one or more lines, or boundaries. The sum of all the boundaries is called the perimeter of the figure, and the portion of surface contained within, is called its area. A circle is a plane figure contained or bounded by a curve line, called the circumference or periphery, which is such that all straight lines drawn from a certain point, within the figure to the circumference are equal to each other. This point is called the centre of the circle, and each of the straight lines is called the radius. The straight line drawn through the centre and terminated at both ends in the circum- c Fig. 7. ference, is called the diameter of the circle. It is plain, from the definition that all the radii must be equal to each other; that all the diameters must be equal to each other, a and that the diameter is always double the radius. In speaking or writing, the circle a is usually denoted by three letters placed, at any distance from each other, around I. the circumference; thus, in fig. 7, the circle is denoted by the letters. A ch, or A E B ; or, by any three of the other letters on the circumference. The point ois the centre, either of the

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straight lines o A, o B, oc, o B, is a radius, and the straight line A B is a diameter. An are of a circle is any part of its circumference; the chord of an are is the straight line which joins its extremities. A segment of a circle is the surface inclosed by an arc and its chord. A sector of a circle is the surface inclosed by an arc, and the two radii drawn from its extremities. Thus, in fig. 7 the portion of the circumference AM c, whose extremities are A and c, is an arc; and the remaining portion A B c, having the same extremities is also an arc; the straight line A c is the chord of either of these arcs. The surface included between the arc A. M. c and its chord A c, is the segment A M c, there is also the segment A B c. The surface included between the radii o c, o B, and the arc c B, is called the sector co B; the remaining portion of the circle is also a sector. A semicircle is the segment whose chord is a diameter. Thus in fig. 7. A c B, or A E B is a semicircle. The term semicircle, which literally means half-a-circle, is restricted in geometry to the segment thus described; but there are many other ways of obtaining half a circle. Plane rectilineal figures are described under various heads; as trilateral or triangular; quadrilateral or quadrangular; and multilateral or polygonal. A triangle (fig. 8) is a plane rectilineal figure contained by three straight lines, which are called its sides. No figure can be formed of two straight lines; hence, an Fig. 8.

angle is not a figure, its legs being unlimited as to length. Triangles are divided into ~ various kinds, according to the relation of their sides or of their angles ; as equilateral, isosceles, and scalene; right-angled, obtuse-angled, and acute-angled. Fig. 9. An equilateral (equal sided) triangle, Fig. 10. is that which has three equal straight lines or sides (fig. 9). An isosceles (equal-legged) triangle, is that which has only two equal sides (fig. 10). A scalene (unequal) triangle, is that which has all its sides unequal (fig. 11). A right-angled triangle is that which has one of Fig.11. its angles a right angle (fig. 11), in which the angle at A is the right angle. The side opposite to the right angle is called the hypotentise (the subtense, or line stretched under the right angle) * and the other two sides are called the base and the perpendicular; the two latter being interchangedble according to the position of the triangle. . - An obtuse-angled triangle is that which has one of its angles an obtuse angle (fig. 8). - - An acute-angled triangle is that which has all its angles acute; figs. 9 and 10 are examples as to the angles, but there is no restriction as to the sides. In any triangle, a straightline drawn from the vertex of one of its angles perpendicular to the opposite side, or to that side produced (that is extended beyond either of its extremites in a continued straight line), is called the perpendicular of the triangle; as in fig. 12, where the dotted line is the perpendicular of the triangle; and in fig. 13, where the dotted line drawn from the point g to the dotted part of the base produced is the perpendicular of the triangle. Fig. 12. A quadrilateral figure, Fig. 13. o or quadrangle, is a plane rectilineal figure contained by four straight lines, called - its sides. The straight = F line which joins the les. is called its di l vertices of any two of its opposite angles, is ca its dugong. Quadrangles * divided into various kinds, according to the relation of their sides and angles; as parallelograms, including the rectangle, the square, the rhombus, and the rhomboid; and trapeziums, including the A Fig. 14. lo trapezoid. A parallelogram is a plane quadrilateral figure, whose opposite sides are parallel; thus, fig. 14, A c B D, is a parallelogram, and A B, co, are its diagonals.

A rectangle is a parallelogram, Fig. 15. whose. angles are right angles (fig. 15). A square is a restangle, whose sides are all equal (fig. M6). A rhomboid is a parallelogram, whose angles are oblique (fig. 14). A rhombus, or lozenge, is a rhomboid, whose sides are all equal (fig. 17). A trapezium is a plane a a quadrilateral figure, whose opposite sides are not parallel (fig. 18). A trapezoid is a plane quadrilateral figure, which has two of its sides parallel (fig. 19). A multilateral figure or polygon, is a plane rectilineal figure, of any number of sides. The term is generally applied to any figure whose sides exceed four in number. Polygons are divided into regular and # r; the former having all their sides and angles equal to each other; and the latter having any variation whatFig.20. ever in these respects. The sum of all the sides of a polygon is called its perimeter,

b - and when viewed in position its contour.

- Irregular Fo are also divided into c convex and non-conver; or, those whose angles are all salient, and those of which

O one or more are re-entrant. The irregular

polygon (fig. 20), has its angles at B, C, and D, salient; and its angles at A and E, re-entrant. Polygons are also divided into classes, according to the -- number of their sides; as, the pen- Fig. 22 *** tagon (fig. 21), having five sides; the g. o. hearagon (fig. 22), having six sides; the heptagon having seven sides; the octagon having eight sides; and so on. According to this nomenclature, the

K.) triangle is called a trigon, and the

quadrangle a tetragon. QUESTIons on THE PRECEDING LEsson.

What is a geometrical solid what is a point? a line? a straight line? a crooked line? a curve 2 How are lines denoted what is a surface? a plane an angle * What are parallel straight lines how are angles denoted? how are they divided ? what is a right angle an oblique angle an acute angle an obtuse angle 7 What is a plane figure ? its perimeter its area a circle its circumference f its centre f its radius * its diameter 2 How is a circle denoted? what is an arc a chord * a segment a sector a semicircle 2 How are plane rectilineal figures divided ? what is a triangle an equilateral triangle an isosceles? a scalene? a right-angled triangle an obtuse-angled? an acute-angled? the perpendicular of a triangle In a right-angled triangle, what are the hypotenuse, the base, and the perpendicular * What is a quadrilateral figure? how are quadrangles divided ? what is a diagonal? a parallelogram a rectangle 7 a square a rhombus * a rhomboid a trapezium ? a trapezoid a polygon? How are polygons divided ? what is the difference between .. and irregular polygons what is perimeter contour * salient * re-entrant? pentagon hexagon?' heptagon trigon? tetragon?

Fig. 18, Fig. 17.

Fig. 19.

- LESSONS IN ENGLISH GRAMMAR.—No. II. ETYMOLOGY. Supposing that you have acquired a pretty good knowledge of the sounds of the different letters in the alphabet, you should next apply yourself to the study of the most correct way of putting them together, so as to form words rightly spelled, and sentences properly constructed. To do this you must find out how words have been formed—how many different sorts of words there are—what is their exact meaning—and the way in which they may be changed or altered, according to the ideas which the writer or speaker may wish to express. This is the soond part of English Grammar, and it is calledEtymology,

Words may be divided into two classes; first, those which are primitive: secondly, those which are derived. Primitives, or original words, are words which have been purposely formed to express one idea; they are words which have not been taken from any other word, and which canno" be reduced so as to be more simple. Thus, the words man, book, child, house, &c., express a complete sense, and nothing can be taken from them. Derivatives are words which are drawn out of others, or which take part or parts of others, or which are formed by joining two words together, so as to make a new word meani something different from the others; as mankind, bookbinder, childlike, housekeeper, &c. All these words you will see may be reduced ; mankind may be reduced to man, and still express a complete sense; and so may the other words. There are ten sorts of words in the English language: these are commonly called PARTs of SPEECH, and the names given to them are as follow :—Article, Noun, Pronoun, Adjective, Werb, Participle, Adverb, Conjunction, Preposition, Interjection. Each of these must be studied separately. That you may have a general view of the whole subject, a brief mention of them will be made here, though each will afterwards be explained more fully. It is supposed that in the English language there are about sirty thousand words; of course, each of these words belongs to one or other of the following ten parts of speech. 1. The ARTICLE is placed before a noun, to point it out, and to fix its exact meaning. 2. The Noun is the name of anything. 3. The PRoNoun is used for, or instead of a noun, to prevent its being repeated too often. 4. The ADJECTIVE expresses the particular quality or property of the noun. 5. The VERB expresses action, being, or suffering. 6. The PARTICIPLE is a word derived from a verb, and partakes of the nature both of a verb and an adjective. 7. The ADVERB describes the quality, or circumstance, or peculiar meaning of other words, and is joined either to a verb, an *::::: a participle, or another adverb. . The CoNJUNction joins the several words or parts of sentences together. 9. The PREPosition is commonly set before words to connect them, or to show their relation. The INTERJECTION expresses some sudden emotion of mind,

I. THE ARTICLE.

THE ARTICLE is a part of speech set before nouns, to point them out, and to fix their exact meaning. There are in the English language two articles, a or an, and the. The a and an are reckoned but as one, because a becomes an when it is placed before a vowel; that is, before a, e, i, o, and u short, as an urn ; also before a mute, or an h that is not sounded, as an hour; if the h be sounded, the article a only is used, as, a haven. The article a is used before all words beginning with a consonant, as, a shoe, a boot; also before u long, that is, when it has the sound of you, as in, a useful book, not an useful book, or, a union, not an union. Articles are divided into two classes, definite and indefinite. An indefinite article is one which does not define the particular meaning or application of the word before which it is placed; or which speaks of things in general, things which are com: mon, or of which there are many of the same sort. A and an are indefinite for this reason; as if you were to say, a book, a man, an apple, you would not be understood as meaning any particular book, or man, or apple. The is called the definite article, because it defines your meaning, and fixes it to one particular thing; as, the book, the man, the apple; that is, some particular É. or man, or apple. If you were to enter a coffee-room, and say to the waiter, “Lend me a paper,” he would bring you the first that came to hand; but if you saw the Times in his hand, and said, lend me the paper,” he would understand you as asking for that particular paper. For this reason the definite article is sometimes called the demonstrative article, as, when pointing to an individual, we say, “That is the man I wish to employ;” or “This is the book I want you to lend me.” The article the may be set before nouns both of the singular and plural number, because we can 3. definitely of many as well as of one; as, the men, the books, the apples.

• The articles are sometimes placed before an adjective, when it precedes a noun, as, an admirable painting, or, The better day the better deed.

The definite article the is sometimes set before adverbs in the comparative and superlative degrees, as, The more the merrier; or, The oftener I look at Raffaelle's paintings, the more I admire them.

Articles are sometimes found joined to proper names, for the purpose of giving them distinction or eminence; thus we say of some large town, it is quite a London, that is, a place as busy or as bustling as London; or, the Howards, that is, the family of the Howards; or, he is a Wellington, that is, a man as distinguished for skill and o as the Duke of Wellington; or, the Caesars, that is, the Roman emperors of the name of Caesar.

Some nouns are used without articles; such as proper names, Andrew, London, Paris; or names of attributed or mental qualities, as, beauty, goodnature, virtue, charity, &c.; or words in which nothing is implied but the mere existence of the thing, as, This is not silk, but cotton; or, This is not gold, but silver gilt. There are also nouns which will not admit the use of the article, as when words are to be taken in their largest and most general sense; thus we say, Man is a rational, an accountable creature; that is, all men, without exception, are rational and accountable.

(Questions on the foregoing Lesson will be given in our next.)

LESSONS IN ARITHMETIC.—No. II.

THE difficulty of inventing names for all numbers even to a limited extent, and of remembering them after they were invented, evidently led to the classification and arrangement exhibited in our system of numeration, which was explained In the first lesson. The next difficulty would be that of performing calculations by the help of the mere names—a process which, in such a case, must either be done mentally, or with the assistance of the ten fingers. The use of smals stones or pebbles (in Latin, calculi), for the purpose of making calculations, is indicated by the origin of the word itself. The necessity of inventing signs or characters to represent numbers, and to facilitate the process of computation, would become more and more obvious as society advanced in civilisation. At first, men would most naturally employ the letters of the alphabet, in every language, as the readiest marks for numbers. Hence, we find that this practice was adopted by the Hebrews, the Greeks, the Romans, and various other nations of antiquity....In the two oldest collections of writings in the world, the Bible, and the works of Homer—the one written in Hebrew and the other in Greek—the letters of the alphabets of these languages are respectively used to denote the whole or parts of these books, in their proper numerical order. The letters of the alphabet in any language, however, would go but a little way in expressing numbers by signs, unless some system of classification and arrangement were adopted, or some method pf increasing their value, according to a fixed scale, introduced. For the purpose of expressing large numbers by signs, the Hebrews divided - the letters of their alphabet into three classes of nine characters each, to denote units, tens, and hundreds, respectively; and as this alphabet ountained only 22 letters, they adopted five of its letters, which had a final form (that is, a peculiar form when they terminated a word), to complete the class of hundreds. The whole collection of signs was then arranged, as in the following table:— HEBREw SystEM of NotATIon. Signs. Names. Walues, Signs. Names. Values.|Signs. Names. Values.

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In §ombining different numbers, the greater is put according to the Hebrew mode of writing Fo .* o S5p, 121: ... The number 15 is marked by the 'letter: Yor Rislead of the letters TT, because the latter commence §: name of God in Hebrew. The thousands are denoted by the

units with two dots above—thus, R, 1,000. Gesenius, in his grammar, says that this numeral use of the letters did not occur in the text of the Old Testament, but was first found On the coins of the Maccabees in the middle of the second cen: tury before the Christian era.

he Greeks, in the same manner as the Hebrews, divided the letters of their alphabet into three classes of nine cha. racters each, to denote units, tens, and hundreds, respectively; and as this alphabet contained only 24 letters, they adopted three marks which were formerly used as letters in the more ancient Greek alphabet, introducing one into the class of units, one into that of tens, and one into that of hundreds. The whole o characters thus arranged are shown in the following able :

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denoted by the same letters with the letter M under—as M

10,000. Instead of placing the letter M under, sometimes the letters Mu were placed to the right of the number of myriads, and sometimes only a point, instead of either, was placed in the same position, to indicate the same thing. Archimedes extended this notation, by taking the square of the myriad as a new unit, or period, and forming a series of periods containing eight figures in each; so that he was enabled to express a number sufficient to denote all the sands of the sea. This system of notation, in some respects, anticipated the modern systems; and in others, surpassed them; but, unfortunately, it was confined to the knowledge of the learned. The same mathematician, one of the mightiest geniuses of antiquity, anticipated the discovery of logarithms, by a few hsppy thoughts, which were allowed to lie dormant for 2,000 years, until Napier promulgated his immortal invention, in 1614, and in his turn forestalled the discoveries of later times.

The origin of the notable improvement in notation, by which the nine, characters, for units only were employed, and the eighteen for tens and hundreds, thrown aside, is still a doubtful question, although it has generally been attributed to tho Arabs, or Moors of Spain, and examples of a similar notation are to be found in India. This improvement consists in giving to these nine characters a relative value—that is, a value depending on their positions, as well as an absolute value depending on their names. Thus denoting the place of units by a certain fixed mark or character, and placing each of the nine primary characters to the left of it, their values are increased tenfold, so that no new characters are necessary to denote the tens ; again, denoting the place of tens and units by two of the same fixed marks or characters, and placing each of the nine

rimary characters to the left of these, their values are increased a hundred-fold, so that no new characters are necessary to denote the hundreds. It is evident that this process may be extended indefinitely, and applied so as to denote not only all the numbers whose names are expressed or indicated in our system of numeration, but all numbers whatsoever, although far beyond the reach of our numerical momenclature. The first nine letters of any alphabet would, of course, answer the purpose of denoting the units, and a dot or any new letter might be employed to denote the vacant places of units, tens,

&c., as they occur. The nine characters, and the cipher

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cipher, one, two, three, four, five, six, seven, eight, nine. M. Chasles, a French writer who has recently made seme curious researches in the history of the mathematics, maintains that these characters have descended to us from the Greeks and the Romans, with the whole system of decimal notation. He asserts that a very obscure passage in the writings of Boethius, a Roman philosopher and senator, who flourished at the end of the fifth . has a direct reference to our decimal system of notation; that various manuscript treatises written between the tenth and twelfth centuries relate to the same subject; that the celebrated Gerbert (afterwards Pope Sylvester II.) had greatly contributed to introduce and extend the knowledge of the decimal system, in the east, towards the end of the tenth century; and that the apparatus employed to facilitate operations in this system, which was unquestionably constructed on the decimal scale, was universally denominated the abacus Pythagoricus, or Pythagoras's board, until the beginning of the twelfth century. In the early part of this century, the cipher, which had been preceded in the Greek notation by a dot or period, was known at first by the names of rota, rotula, sipos, and afterwards by those of circulus and cifra, or cipkra. The term cipher, however, is applicable to all the characters used in our system of notation; and the art of arithmetic itself is hence called ciphering. This word is evidently derived from the Hebrew verb saphar, to number; and it is not improbable that even the art of numbering itself, as well as the symbols employed in it, may have * over the east from the people who originally spake is language. At the revival of literature, the invention of the arithmetical characters was ascribed to the Indians, from whom it was pretended they came to us, through the Arabians; and the characters themselves were denominated figura, Indorum. In this way all trace of the ancient system of notation, preserved in the abacus Pythagoricus, was insensibly lost in the writings cf the moderns, while some ideas, no doubt taken from the Arabic literature, were introduced. Hence, at the present time, all remembrance of the abacus, and of the real origin of our system of notation, has disappeared, and their origin is referred to the Arabians and the Hindoos, Many passages, however, in the works of writers on the subject, even so late as the sixteenth century, show that the Greek and Latin origin of our decimalsystem was not then completely forgotten. It is also important to observe, that those who ascribe its origin to the Arabians, or the Asiatics, do not assert that we have preserved the characters employed by the inventors; and it may, indeed, be shown that the characters we now use are extremely analogous to those of Boethius, as well as to those which were employed in the treatises on the abacus written during the middle ages. ... The following are the characters which M. Chasles has discovered in a manuscript

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From the preceding observations, it is evident that the Romans, especially those who were initiated into the doctrines of Pythagoras, employed in their mathematical calculations a system of notation and a set of characters very similar to our own, and very different from those which they used in their ordinary writings, and which are denominated Roman figures. As these are, to this day, very often used for various common purposes among ourselves, such as paging certain portions of a book, marking dates, numbering chapters, &c., it will be useful to give the following table of their values:—

WULGAR SystEM or Roman NoTATIon.

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121, &c. The thousands are denoted by the letter M or the combination CIO; the tens of thousands by CCI00; and so on. In this system, there is very considerable regularity and . although it is not adapted for the purposes of calculation. The letter I, repeated any number of times denotes so many units ; when placed to the right of another character, it adds a unit to the number represented by that character; when placed to the left, it takes away a unit. The letter X repeated any number of times, denotes so many tens; when placed to the right of a character of greater value, it adds ten to the number represented by that character; when placed to the left, it takes away ten. }.he letter C repeated any number of times, denotes so many hundreds; when placed to the right of a character of greater value, it adds a hundred; and when placed to the left, it takes away a hundred; and so on, with M &c. The letter C placed right and left, in the former case inverted, increases the value of CIO tenfold. A bar placed over any character or number increases its value a thousand fold. some few cases, it has been ascertained that the Romans employed these literal characters even with values depending on their position ; thus, in Pliny, we find XVI.XX,DCCC.XXIX used for the number 1,620,829.

The following tables, especially the first, which combines the system of numeration with that of notation, will be found of the greatest utility.

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1, 00 0, 00 0, 00 0, 00 0, 00 0, 00 0, 00 0, 00 0. The use of these tables is to enable the learner to read any number that may be placed before him, or that may occur in calculation, or to write any number that may be proposed in words, with ease and certainty, ...; to the particular system he chooses to adopt, or that may be recommended to him. As we are partial to the English system, on account of its greater ability to grasp large numbers, we add the following rules, which have special reference to Table I., and which must be committed to memory:— Pnoblox. 1.--To read or express in words any proposed number expressed in figures.—Rule: #. the proposed number off by commas into P. of six figures each, from right to left, and then into hal Fo of three figures each. Apply to the number thus divided the names which belong to the respective figures and places of figures in each period, reading them from left to right—that is, from the highest name to the lowest—taking care to avoid repetitions of the same word, and observing that wherever a cipher occurs, the name which belongs to that place must be omitted. It will be of importance to remember, that in the application of this table, the name hundred is applied to that of every third figure, and the additional name thousand to that of every sixth figure, reckoned from right to left. After the name hundred, in reading from left to right, the name ten immediately follows; and after the name hundred thousand, that of ten thousand, in the same order. After the name ten, in reading from left to right, the name of the period in which it occurs is applied; and after the name ten thousand, that of thousand is applied. The following examples will show the application of this rule. ExAMPLE 1.-Read, or express in words, the number 146385297831276543 Here, the number pointed according to the rule will stand thus: 146,385;297,831;276,543. . . In this number there are three complete periods, and six half periods. By referring to the table, we see that the name of the first period, reckoning from left to right, is billions, the next “millions, and the next units. - Hence, remembering the above rule, it is read or expressed in words, thus: One hundred and forty-six thousand three hundred and eighty-five billions, two hundred and ninety-seven thousand eight hundred and thirty-one millions, two hundred and seventy-six thousand five hundred and forty-three. The name units is generally omitted at the end of a number when read in the preceding manner. The number in the table itself is read thus : one septillion, ExAMPLE 2.-Read or express in words the number 101.000230200001. This number, pointed according to the rule, will stand thus: 101;000,230;200,001. In this number there are two periods and a half; hence, it is read or expressed in words as follows: One hundred and one billions, two hundred and thirty millions, two hundred thousand and one. PnoRLEM 2.--To write or express in res any proposed num ber expressed in words.-Rule *. o line 3. many ciphers, from right to left, as will extend from units to the highest name in the proposed number, and point them off according to the table, as before, Write below these ciphers the figures which express the

names that are applied to each place in the given number, according as they stand in the table, and fill up the blank places in this line of figures with ciphers; then the proposed number will be properly expressed in figures. ExAMPLE 1.-Write or express in figures the number; Three hundred and forty-five thousand siz hundred and seventy-two bil lions, one hundred and thirty-eight thousand seven hundred and ninety-two millions, five hundred and eighty-three thousand six hundred and forty-one. Here, the highest name in the proposed number being hundreds of thousands of billions, we write eighteen ciphers, as follows; and then write below them the figures which express the names belonging to each place, according to rule; thus: 000,000;000,000;000,000. 345,672; 138,792;583,641. ExAMPLE 2.-Write or express in figures, the number; One ... billions, two thousand and thirty-two millions, one hundred area. ooze.

Here, as before, the highest name being hundreds of billions, we write fifteen ciphers as follows; and then write below them the figures which express the names belonging to each place, according to rule, filling up the blank places with ciphers; thus:–

000;000,000;000,000. 100;002,032;000,101. QUESTIons on the PRECEDING Lesson.

1. Write out the names of all the numbers from one to a hundred, and express them in figures.

2. Write out the names of the numbers which immediately follow one hundred; one hundred and ninety-nine; four hundred and ninety-nine; nine thousand nine hundred and ninety-nine; and a million.

3. Express, in figures, the numbers named in the preceding example, and those which immediately follow them.

4. Write the names of the numbers which are next to the following numbers, and oil." both sets in figures: one million and ninety-nine; one million five thousand nine hundred and ninetynine; and nine millions nine hundred and ninety-nine thousand nine hundred and ninety-nine.

5. Read or express the following numbers in words:—

202 20030208 100010001000

1001 1010101 . 3000000000000 15608 99.99999 7776665.55444 306042 347125783 123456789123 5678914 202021010 4848484.8484.848 26312478 9090909090 10210230430400

6. Write or express the following numbers in figures — Four hundred and four. Three thousand and thirty-two. Twenty-four thousand and eighty-six. Six hundred and five thousand, and nineteen. Eleven thousand, eleven hundred and eleven. Three hundred and forty-one thousand, seven hundred and eighty-two. Eighty millions, two hundred and three thousand and two. Two hundred and two millions, twenty thousand two hundred and two. Nine thousand nine hundred and ninety-nine millions, nine hundred and ninety-nine thousand, nine hundred and ninety-nine. Write also the number which follows this last one in order Twenty thousand millions. Two hundred thousand and twenty millions, two thousand. One trillion. The next number to thirty thousand billions, ninety-nine thousand.

nine hundred and

LESS ON S IN FR EN CH-No. II. By Professor Louis FAsque LLE, LL.D. The following reading and pronouncing exercises belong to Lesson I., given in our last number, and must be carefully erformed in connexion with that lesson, which should have een headed Section I. for the sake of reference. ExEncise 1.--THE Wowels. (a) Table, table; fable, fable; chat, cat; Éclat, splendours. arbre, tree; tard, late; balle, ball. (a) ame, soul; blame, blame ; bâtir, to build; pâte, paste; age, age; mat, mast. (e) me, me; de, of que, that ; elle, she 3 malle, mail; parle, speak; fourchette, fork; salle, hall.

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