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heaven in a chariot of fire | The records of the martyr age are filled with similar instances. These men dignify humanity; “they were stoned, they were sawn asunder, were tempted, were slain with the sword: they wandered about in sheepskins and goatskins, being destitute, afflicted, tormented; they wandered in deserts and in mountains, and in dens and caves of the earth,”— but their moral courage forsook them not; their souls were strong to a high purpose; they made the dungeon vocal with songs of gladness, and sealed their principles with their blood How greatly does all earthly majesty pale before these illustrious examples of moral grandeur ! We rejoice, indeed, that we are

not called upon to defend our principles in the midst of such Violence; let us, nevertheless, understand these principles, and never prove treacherous to them; we are called upon to work them out in life; the eyes of men are continually upon us, and every action is compared with the professed principle, and the lynxeye of an opposing or malicious world will not be slow to detect eyery discrepancy, and its tongue to tell every failing with clamorous pertinacity. ... If conduct be blameless, conscience will be approving, and the testimony of a good conscience will not fail to oonsole and cheer amid the din of violent opposition, or the power of a secret conspiracy.

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Irregular Verbs continued.

(3) stonnen, to be able. (See Remark 10.)

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INDICATIVE. StJ BJ UNCTIVE. CON DITIONAL. IMPERATIVE* | IN FINITIVE. | PARTICIPLE. Present Tense. Present Tense. Wanting. | Present Tensel Present. ; ( lid fann, I am able. id) finne, I may tonnen, to be fünment, being % 3. bu fannīt, thou art able. |bu fümnest, thou mayst 6 able. able. 3 (3}et faith, he is able. er fönne, he may 2 a: (1|moir fūnnen, we are able. |mit fönnen, we may GD E {2|iffe füntet, you are able. |ifft fönnet, you may so * {3|sie fênmen, they are able. |sie fünnen, they may Imperfect Tense. Imperfect Tense. ; (1jić tomte, I was able. , , is tomte, I might . § {2|bit fomites, thou wast able.|tit femtest, thoumightst go 3, ( 3 er founte, he was able. |er fönnte, he might 3' 5 (1swit femten, we were able. |pitfonaten, we might Q) § {2|ist fountet, you were able. ||ist fönntet, you might a. * (3|sie femitten, they were able |fic finisten, they might Perfect Tense. Perfect Tense. Perfect Tense. Perfect. id) safe gefonnt, I have been|id sq6e gefonnt, I may have fplint bu hast gefonnt, able, &c bu ()(ibeft Jefonnt, been able, 9°fount haben, gefonnt, been w y 2 to have been able er Wat gefonnt, cr ()ābe gefonnt, &c. able.

First Future.

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ON PREPARING SHELLS.

In general it happens that, when shells become dry, they lose imuch of their natural lustre. This may be very easily restored, by washing them with a little water, in which a small portion of gum arabic has been dissolved, or with the white of an egg. This is the simplest of those processes which are employed, not only by the mere collector, but by the scientific conchologist. There are many shells which have a very plain appearance on the outside, by reason of a dull epidermis or skin with which they are covered. This is removed by soaking the shell in warm water, and then rubbing it off with a brush. When the epidermis is thick, it is necessary to mix with the water a small

portion of nitric acid, which, by dissolving a part of the shell, destroys the cohesion of the epidermis. is acid must be employed with great caution, as it removes the lustre from all the parts exposed to its influence. The new surface must be polished with leather and tripoli. But in many cases even these methods are ineffectual, and the file and the pumice-stone must be resorted to, in order to rub off the coarse external layers which conceal the beauties of the shell. Much address and experience are necessary in the successful employment of this process. Their reward, however, is often great. When thus prepared by the artist, even the common mussel is most beautiful,

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Xopia: We cannot undertake to correct exercises.

ZETA (Windsor): The vocabularies need only be learnt one after the other, as they are wanted, not altogether. W. R. C. : You are right in your supposition. PHILOMATH : The word is intended for eart. You mistranslate the other sentence, which should be rendered thus: “Dishonour follows wickedness.” The article of the noun in the genitive may sometimes be omitted, but should generally be used when the English definite article occurs. The Jetter y is never pronounced soft. Lessons in elocution will shortly appear. J. R. H. : The first Lesson in Chemistry is contained in Part XVI., No. 7&, p. 259, vol. iii.-D. McK. (Row): Water can never rise higher than its level in any system of pipes, unless the forcing-pump be employed.—EIN JUSGER IRLANDER: It is impossible to tell which country has produced the most eminent men as compared with the population, unless you sit doggedly down, and with the census of a given period in the one hand, and a list of the eminent men who have flourished within that period in the other, compare the numbers of the former with the numbers of the latter, for England, Scotland, and Ireland. Looking at the smallness of the population, and the great number of eminent men who have arisen in Scotland, we guess, Rut we only guess, that the result of the comparison would be in favour of that country. Power-Looms WEAVER : By all means study our Lessons in Geometry in the P. E.; they will make a man of you.—QUAESITOR ANXIUS (Jersey): Your friends are right about the expense of becoming a civil engineer. To give you an outline of the requisite studies would fill a volume. The

best outline we have seen is contained in the programmes of the “Ecole ||

Centrale des Arts et Manufactures,” &c., at Paris: 18. Programme des Connaissance exigée pour admission à l’Ecole Centrale; 2°. Programme des

Cours, première Année ; Deuxième et Troisième Année, &c.—AMATEUR : ],

Thanks for his sketch of a portable laboratory; the first chemists have liad only a few things collected on an old tea-tray.—GEORGE IR. (Old l’ish-street): See Dr. Beard’s Lessons in English in the P. E.

F. F. H. ENBEST (Fordingbridge): We shall be most happy to be favoured with a further explanation of his ideas on the motions of the earth and the *moon, either by diagram or machinery; an annual rotatory of the earth on an axis perpendicular to the plane of its orbit, and a new explanation of the amoon’s monthly rotation, are certainly novelties worthy of our attention. PHILo (Nottingham): His corrections on the Key to Cassell's Arithgnetic are all right; a few of the answers in the Key were taken from the answers published by the original authors of the work, trusting that they were correct ; but whether the errors he has discovered arose from misprints, or carelessness in the calculators, it is difficult now to say,+ W. WALKER (Southport) : In the last edition of “ Keith on the Globes,” in 1851, you will find the number of the constellations stated at 96, as taken from the Royal Astronomical Society’s catalogue; and 9 more are added as given by foreign mathematicians. \ W. Jones (Stockport): The Lessons in Arithmetic are not completed; your writing is rather stiff, and your spelling is very bad.—R. BLONELEY: His suggestion as to training-schools will be borne in mind.—J. MARSHALL (Suffield): Hymer’s “Physical Astronomy;” but the grand book is Laplace's * Mecanique Celeste; ” there is a collection of tables called the “Requisite Tables” to be used with the “Nautical Almanac.”—AN OLD SUBSCRIBER (Shrewsbury): We should be glad to please him by inserting his letter; but it is contrary to our rule to lend our pages to puff or praise any works of which we do not know the value. W. W. W. : To the exercise appended to Prop. II., Book IV. of Cassell's Euclid, add the words “only in the case of the equilateral triangle.”— John DAVIES (Pontypool) should get a copy of Cassell’s “Emigrant's Almanac,” price 6d. f S. GENT (Sutton-in-Ashfield): His method of calculating discount is no doubt the true one ; but it is never used in practice; we must therefore follow the customary method, which we have done.—A. P. B. (Hoxton): The best proof of elasticity is the rebound of the elastic substance; but according to his theory clay would be elastic.—S. G. J. (Drogheda): The study of two languages together is very likely to lead to confusion; we would therefore recommend that of a language and a science together.—W. F. P. (Ilondon): “A farmer had £100 to buy oxen, sheep, and geese, so that he should have 100 animals in all, the oxen being £5 a-piece, the sheep £1 a-piece, and the geese ls. a-piece. IIow many of each did he buy 2 Here, let w, y, and z denote the numbers of each sort of animal; then by the question we have these two equations :-(1-) &-Hy-H2=100, and (2) 100a;+20y--z=2000; subtracting (l.) from (2.) we have 992--19y=1900, 1900–992: 9

and therefore 2: = = 100–5* + +: now, as the e can be no

fractions, # 4a must be divisible by 19; and therefore ar-19; whence y=l, and z=80° The answer is, therefore, 19 oxen, 1 sheep, and 80 geese. The other question is useless, and so plain that it needs no explanation. J. E. (Kidderminster): Thanks for his corrections; if any sections seem to be omitted, it can only be by mistakes in the mnmbering.—GAUTIER (Bristol): “Uncle Tom's Cabin" in French, and the “Key” to the same, may be obtained at our office.—P. H. : We would recommend the study of French and Latin together to those who have time at their disposal.-A READER in Ulvertone, whose signature we cannot read, is referred to Messrs. Horne, Thornthwaite and Co., 123, Newgate-street, for the price of daguer

must be a whole number; in order that this may be the case,

o apparatus. We would scarcely advise any one to emigrate to aytl. IR. SIMöN His plan is excellent.—WM. Jos Es should study the Jessons from the W. M. F. first.—R. W. (Greenwich): Right.—J. BATEs (Halifax) is Wrong about the formula to which he refers at p. 103, vol. iii., and we are right.-GEORGE DEWDNEY (Red-hill): His solution of the Wolf question was algebraic and elegant, but we have not room for it.—AN IGNorANT BoLTONIAN asks whether it be lawful for a lady to marry her father's latybrother; we do not know about halves, but we will give him, and such of our readers as require it, a list of the wholes, and we think that they include the halves; this list is taken verbatim from an edition of the Bible, printed and published by royal authority at Edinburgh, in the year that Queen Victoria was crowned (1838):— “A TABLE OF KIND RED AND AFFINITY, wherein whosoever are related are forbidden in Scripture, and by our Laws, to marry together.

A man may not marry his A woman may not marry her

1. Grandmother 1. Grandfather

2. Grandfather's wife 2. Grandmother's husband 3. Wife's grandmother 3: Husband’s grandfather 4. Father’s sister 4. Father’s brother 5. Mother's sister 5. Mother’s brother 6. Father’s brother’s wife 6. Father's sister's husband 7. Mother’s brother’s wife 7. Mother’s sister’s husband .8. Wife's father's sister 8. Husband’s father’s brotjuer 9. Wife’s mother’s sister 9. Husband’s mother’s brother 10. Mother 10. Father

ll. Step-mother ll. Step-father

12. Husband’s father 13. Son 14. Husband’s son

. Wife’s mother I}aughter 14. Wife's daughter

15. Son's wife 15. Daughter's husband 16. Sister 16. Brother 17. Wife's sister 17. Husband’s brother 18. Brother’s wife 18. Sister’s husband

19. Son’s daughter 19. Son’s son 30. Daughter's daughter 20. Daughter's son 21. Son’s son’s wife 21. Son’s daughter’s husband 22. Daughter’s son’s wife 22. Daughter's daughter’s husband

. Wife's son’s daughter 23. Husband’s son’s son

24. Wife's daughter's daughter 24. Husband’s daughter’s son 25. Brother’s daughter 25. Brother’s son 26. Sister’s daughter 26. Sister’s son 27. Brother’s son’s wife 27. Brother’s daughter's husband

28. Sister’s daughter’s husband 29. Husband’s brother’s son 30. Husband’s sister’s son

. Sister's son's wife 29. Wife's brother's daughter 30. Wife's sister's daughter

L IT E R A R Y IN OTI C E S.

t FRENCH. . CASSELL’s FRENGH DICTIONARY, in Numbers, 3d, each; Parts, ls. each. The French-English division now ready, in stiff covers 4s, or strong cloth 5s. The entire work will shortly be ready, price 8s. 6d. strongly bound. CASSELL’s. COMPLETE MANUAL OF THE FRENCH LANGUAGE. By Professor De Lolme. Price 3s. in neat cloth. CASSELL’s LEssons IN FRENCH. Part I. By Professor Fasquelle. Price 2s. in stiff covers, or 2s. 6d. meat cloth. Part II. will be ready in a few days CASSELL’s KEY TO THE ABOVE LEssons, 1s. in paper covers, or \s. 6d. neat cloth. CASSELL’s SERIES OF LEssons IN FRENCH, on an entirely novel and simple plan. Price 6d., or per post 7d. ar , r - LATIN. CASSELL’s GRAMMAR OF THE LATIN LANGUAGE. By Professors An. and Stoddard. Revised and corrected, price 3s.6d., will shortly be ready. * CASSELL’s LATIN DICTIONARY. By J. R. Beard, D.D., will be issued in Weekly Numbers at 3d., immediately on the completion of the French Dictionary. The entire price of the volume, bound, will be 8s. 6d. osaurs LESSONS IN LATIN. Price 2s. 6d. paper covers, or 3s. neat ClOth. * * CASSELL's KEY. To THE ABOVE LEssons. Price Is... in paper covers, or ls. 6d. neat cloth. CASSELL’s FIRST LESSONS IN IATIN. By Drs. Andrews and Stoddard. Price ls. paper covers, or ls. 6d. neat cloth. - GERMAN. t CASSELL's GERMAN DICTIONARY is now issuing in Numbers, at 3d. each Monthly Parts, ls, each. losaurs LEssons IN GERMAN, price 2s. in stiff covers, or 2s. 6d. cloth.

CASSELL’s CLASSICAL, LIBRARY, containing a Latin Reader, with Easy Fables. Vol. I., price ls. 6d. in meat cloth, is now ready. Vol. II. is in course of issue in Weekly Numbers, at 2d., cach, cousisting of Latin Exercises intended to ground the student in the Syntax of the Language. Vol. III. will contain the Acts of the Apostles in Classical Greek, with Notes, and a Lexicon explaining the meaning of every word—the whole carefully revised and corrected.

CASSELL’s EUCLID.—THE ELEMENTs of GeoMETRY. Containing the First Six, and the Eleventh and Twelfth Books of Euclid. Edited by Professor Wallace, A.M., price ls. in stiff covers, or ls. 6d. meat cloth, –KEY. 3d.

CASSELL’s ELEMENTS OF ALGEBRA (uniform yrith Caasall's EUCLID), price ls. in stiff coyers, or ls. 6d, neat cloth,

INSTRUMENTAL ARITHMETIC.-No. III.
T H E PLAN E S C A LE AN ID P R O T R A CTO R.
(Continued from page 13.)

In order to give our students some idea of the other lines drawn on the Plane Scale, we must explain some of the terms employed in Trigonometry. The definition of an angle has been given in the Lessons on Geometry; but in Trigonometry this définition is greatly altered and extended. Angular magnitude in general is the space generated by the revolution of a straight line about one of its extremities which remains fixed; and an angle is the space between the initial and terminal positions of the straight line, whatever be the quantity of revolution. Thus, in fig. 1, let oa be a straight line which revolves about the fixed extremity o, and let o A be its initial position in general; then, if o M be its first terminal position, a o M is an angle in what is called the first quadrant, and is less than a right angle.

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In order to explain the different quadrants, it will be suffi-
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lent to remark, that by the 2nd corollary to Prog XIII., |greater than one right angle; if the terminal position of o A. assell's Euclid, all the angles about a point are equal to four lie in the third quadrant, as o N’, the angle is said to be ght angles; and if A A' and B B' be two straight lines drawn greater than two right angles, and less than three right

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it lies in the fifth quadrant, and the angle is said to be greater than four right angles, and less than five right angles; and S{) OIl. Angles are in practice measured by the arcs of circles intercepted between the initial and terminal positions of the revolving straight line. The circumference of every circle is by convention divided into 360 equal parts called degrees, and marked *; for minuter parts, the degree is divided into 60 equal parts, called minutes; for second minuter parts, the minute is divided into 60 equal parts called seconds; this is called the sexagesimal (from Lat. Sexagesimus, the sixtieth) division of the circle. In France they adopt the centesimal division of the circle, but it has its disadvantages. In the sexagesimal division, it is plain that a right angle is measured by a quadrant of the circle, hence it is said to contain 90° ; two right angles are measured by two quadrants of a circle, or by a semicircle, and are said to contain 180° ; three right angles are measured by three quadrants of a circle, and are said to contain 270° ; and four right angles are measured by a complete circle, and are said to contain 360°. Moreover, five right angles are measured by five quadrants, and are said to contain 450° ; and so on.

The instrument used for measuring angular space, or angles in general, is called the Protractor, as shewn in fig. 2, and consists of a brass semicircle graduated (that is, marked with degrees) from 0° to 180° either way, so that every are has its supplement marked along side of it, there being two rows of Tumbers, one from right to left, and one from left to right; that is, as in the figure, 0° begins at A, and 10°, 20°, 30°, &c., are marked on the outer edge of the instrument, terminating at B, which is marked 180°; and again 0° begins at B, and 10°, 20°, 30°, &c., are marked on the inner edge of the instrument, terminating at A, which is marked 180°. The use of this instrument is to protract (Lat. protraho, to draw out), that is, to lay down an angle of any given number of degrees; it is also employed to measure any given angle, that is, to ascertain the mamber of degrees which a given angle contains. Besides the semicircular form of the Protractor, there is another form which is sometimes put on an ivory Plane Scale, as that within the semicircle in fig. 2; and sometimes on an ivory Parallel Ruler, as in fig. 3. This form of the parallel Tuler, although old, is in our opinion one of the most conYenient; it appeared in the mathematical instrument makers' shops in London about 1760. The graduations of the Semicircular Protractor are transferred to the Scale Protractor, which is marked and numbered in the same manner as the former, by placing a ruler or straight edge on the centre C, and on the several divisions of the semicircumference A D B in succession; then marking on E F, the edge of the scale, the intersections of the straight edge with that edge by portions of the straight limes that would be drawn from C to each division of the semicircumference. The diameter A B of the semicircle is called the blank edge of the Semicircular Protractor, and part of it, as in the figure, is the blankedge of the ScaleIProtractor. Thus you see that in the latter, three of its edges are occupied with the transferred graduations of the semicircle, and the fourth edge is blank, but contains the centre of the semicircle C, which is at an equal distance from the extremities of the scale. Although we have combined these instruments in the figure, to show their construction, the student is not therefore to suppose that there is any such combined instrument in use ; either will serve the same purpose, but one of them is enough in a case of instruments; the Semicircular Protractor is perhaps more handy in practice, but the Scale Protractor is more portable, and in some cases is more useful than the former. The graduated edges of both these instruments should be bevelled almost to sharpness, in order to admit of the easier pointing off of the divisions or degrees of the limb, the name applied to the graduated edge. In some cases, Protractors are made completely circular; and for many purposes they are highly Taseful, especially for laying down plans in surveying. In Trigonometry, which originally meant the measuring of triangles, there are certain straight lines drawn in and about an angle, whose ratios to one another are called Trigonometrical Functions, or simply Trigonometrical Ratios. Referring to fig, 3, if we take the angle A o M in the first quadrant, and from

over o A in its revolution, and reached it a second time, then

the angular point ocut off a certain part o M of the revolving straight line which generates the angle A o M, and from the point M draw M P perpendicular to the straight line o A, its initial position, we shall then form what is called the Elementary Triangle o M. P. In this triangle, which by construction is right-angled, the ratio of the perpendicular M P to the hypotenuse om, is called the sINE of the angle A ox; the ratio of the perpendicular M P to the base o P is called the TANGENT of the angle A o M ; and the ratio of the hypotenuse o M to the base o P is called the SECANT of the angle Ao M. Again, the ratio of the base o P to the hypotenuse om, is called the costNE of the angle A o M ; the ratio of the base o P to the perpendicular MP is called the COTANGENT of the angle A o M ; and the ratio of the hypotenuse o M to the perpendicular M P is called the coseCANT of the angle A o M. The latter three ratios might have been defined as the reciprocals of the former three ratios; thus, the cosine is the reciprocal of the secant; the cotangent is the reciprocal of the tangent; and the cosecant is the reciprocal of the sine.

Referring to fig. 1 again, if we take the angle A o M' in the second quadrant, and from the angular point o cut off a certain part o M' of the revolving straight line which generates the angle A o M', and from the point M' draw M'P' perpendicular to the initial straight lime o A, produced to A', we shall then form the Elementary Triangle o M'P' for the ratios belonging to the angle A o M'. Here, as in the preceding case, the ratio of the pendicular M'P' to the hypotenuse o M' is the SINE of the angle A o M'; the ratio of the perpendicular M'P' to the base o P’ is the TANGENT of the angle A o M' ; and the ratio of the hypotenuse o M' to the base o P' is the SECANT of the angle Ao M'. Also, the ratio of the base o P' to the hypotentise o M' is the cosin E of the angle A o M' ; the ratio of the base o P' to the perpendicular M'P' is the cotANGENT of the angle Ao M'; and the ratio of the hypotenuse o M' to the perpendicular. M'P' is the coseCANT of the angle A o M'.

The trigonometrical ratios belonging to the angles A o M and A o M' above explained, are usually exhibited in the following abridged forms:–

Pirst Quadrant.

* — sine W OM MP Top = tangent O OM E Secant ! OIP of the angle A o M. 9* = cosine OM OP = cotangent IMP OM = COSecant MP * Second Quadrant. M'P' e M. P. - = Sime Y OM f_ A ** = tangent OP f ow E Secant OIP of the angle A o M'. y or = cosine OM op' — Toy - cotangent f OM —or = cosecant M. P -d

The usual contractions for these and some other ratios are given in the following table:–

Names of Ratios, Contractions. sine sin. tangent tan. Secallt SeC. cosine COS,

*** *------ *** - a

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