heaven in à chariot of fire! The records of the martyr age are not called upon to defend our principles in the midst of such filled with similar instances. These men dignify humanity; violence ; let us, nevertheless, understand these principles, and "they were stoned, they were sawn asunder, were tempted, were never prove treacherous to them ; we are called upon to work slain with the sword: they wandered about in sheepskins and them out in life; the eyes of men are continually upon us, and patskins, being destitute, afflicted, tormented; they wandered in every action is compared with the professed principle, and the lynxdeserts and in mountains, and in dens and caves of the earth," eye of an opposing or malicious world will not be slow to detect but their moral courage forsook them not; their souls were strong every disorepancy, and its tongue to tell every failing with to a high purpose; they made the dungeon vocal with songs of clamorous pertinacity. If conduct be blameless, conscience will be gladness, and sealed their principles with their blood ! How approving, and the testimony of a good conscience will not fail to greatly does all earthly majesty pale before these illustrious console and cheer amid the din of violent opposition, or the power examples of moral grandeur? We rejoice, indeed, that we are of a secret conspiracy. LESSONS IN GERMAN.-No. LXXI. Irregular Verbs, continued from p. 76. (2) Dürfen, to be permitted, to dare. (See Remark 9.) PARTICIPLE. Present, I am I may dürfend, being PLUR. SING. permitted. we are be permitted. PLUR: SING. permitted. be permitted. ich dars, ich dürfe, 2 Du darfst, thou art bu dürfeft, er darí, he is er diirfe, wir dürfen, wir dürfen, 2 ihr dürfet, you are ifr dürfet, you may fie dürfen, they are sie dürfent, they may Iinperfect Tense. Imperfect Tense. ich durfte, I was ich dürfte, I might 21 du durfteft, thou wast ou dirstest , thou mightst 3 er durfte, he was er dürfte, he might wir durften, we were wir dürften, we might 2 ist durftet, you were ihr dürftet, you might fie dürften, they were sie dürften, they might Perfect Tense. Perfect Tense. 1 ich habe I have ich Babe I may have 2 ou haft thou hast du habest been permit3 er hat he has er babe ted, &c. 1 wir haben wir haben 2 ihr habet ihr habet 3 sie haben fie haben Pluperfect Tense. Pluperfect Tense. i ich hatte I had ich hätte I might have 2 du hattest thou hadst du hättest been permit 31 et hatte he had er hätte ted, &c. 1 wir hatten wir hatten ihr hattet ihr hättet fte vatten sie hätten First Future Tense. First Future Tense. First Future. Ilich werde I shall ich werde (if) I shall be ich würde 21 du wirst thou wilt bu werbeft permitted, su würdest 31 er viro he will &c. er würde 1 wit werben we shall wir würden ihr würdet 3 sie werden sie werden fte njürden Second Future Tense. Second Future Tense. Second Future. ich werde I shall have ich werde (if) I shall ich würde 2 Du wirst been per- du werdest have been du würdest 31 er wird mitted, &c. er werde permitted, ler würde 1 wir werden &c. wir werben wir würden 21 ihr werbet ihr werdet ihr würdet 3/fie werben fie werben lite würden (9) Remarks on bürfen. sein, it may or might be too late now: Es dürfte vielleicht wahr sein, This verb is commonly rendered to dare, though the primary it might perchance be true. It also signifies, to need, to have sense seems to be that given above, viz., to be permitted: the sig- occasion, &c.; as, Er darf nur reden, he needs only speak; Er nification, to dare, is one in which it is seldom used. The verb darf sich darüber nicht wundern, he must not or should not wonder at is also employed (only in the Imperfect Subjunctive, however), that. When used without an infinitive after it, one must be to denote what probably may be, and may then be translated by supplied to complete the construction: thus: &r darf nicht in das such words as might, need, would, &c.: thus, && vürfte jeßt zu spät | Haus (tommen), he ventures not (to come) into the house. Irregular Verbs continued. (3) Können; to be able. (See Remark 10.) PARTICIPLE. Wantiry. Present Tense. Present. können, to be fönnend, being able. able. I may he may we may be able. PLUR. SING. be able. PLUR. SING. Present Tense. Present Tense. I am able. ich fönne, thou mayst wir können, you may fie fönnen, they are able. sie können, they may Imperfect Tense. Imperfect Tense. I might he inight wirtinten, we might Perfect Tense. able, &c du babest gekonnt, been able, cr habe gefonnt, &c. 1 wir haben gekonnt, wir haben gekonnt, 2 ihr habet gefontit, ihr habet gefonnt, 3) fic haben géfonnt, sie haben gekonnt, Pluperfect Tense. Pluperfect Tense. ich Hátte gefonnt, I had been ich hätte gefonnt, I might have 21 du hattest geforint, able, &c. ou hättest gekonnt, been able, 3 cr hatte gekonnt, er hätte gekonnt, .&c. wir hatten gekonnt, wir hätten gefonnt, 2 ihr hattet gefonnt, ihr hättet gefonnt, fte Batten gekonnt, sie hätten gefonnt, First Future Tense. First Future Tense. First Future. ich werde können, I shall be ich werde fönnen, (if) I shall ich würde du wirft können, áble, &c. but werdeft fönnen, be able, tu würdest 3 er wird könneti, er werbc fönnen, &c. er würde 1 wir werben tönnen, wir werten können, wir würden 2 ihr werdet fönnen, ihr werdet fönnen, ihr würdet 31 fic werden fönnen, fie werden können, fie würten Second Future Tense. Second Future Tense. Second Friture. I have been ich werte (if) I shall ich würde 2011 wirst able, &c. tu werdest have been su húūrtest 3 er wird ct wethe able, &c. er würde ri wir werben mir werden wir würden 21 ihr werdet ihr werdet ihr fürbet3 ste werden fle werden sie würden PLUR. SING. PLUR. SING. finncity able, &c. (10) Remarks on können. Er kann es verstanden haben, he may (possibly) have understood it. The original signification of fönnen was to know, or to know It differs, therefore, from dürfen, when it (bürfen) is used (in the how; hence the present sense, to be at liberty to do a thing, to Imperfect Subjunctive) to express possibility ; for fürfen not be able; as, ich fann refert und fchreiben, I can (kràow how to) read only signifies that the thing may be, but that it probably is or and write. Its chief power now is to indicate bare possibility, will be. Können, like dürfen, has sometimes an infinitive underand hence it is often aptly translated by the English, may : as, I stood after it to complete the construction. ON PREPARING SHELLS. portion of nitric acid, which, by dissolving a part of the shell, In general it happens that, when shells become dry, they lose destroys the cohesion of the epidermis. This acid must be much of their natural lustre. This may be very easily restored, employed with great caution, as it removes the lustre from all by 'washing them with a little water, in which a small portion the parts exposed to its influence. The new surface must be ' arabic has been dissolved, or with the white of an egg. This is the simplest of those processes which are employed, not these methods are ineffectual, and the file and the pumice-stone only by the mere collector, but by the scientific conchologist. must be resorted to, in order to rub off the coarse external There are many shells which have a very plain appearance on layers which conceal the beauties of the shell. Much address the outside, by reason of a dull epidermis or skin with which and experience are necessary in the successful employment of they are covered. This is removed by soaking the shell in this process. Their reward, however, is often great. When warm water, and then rubbing it off with a brush. When the thus prepared by the artist, even the common mussel is most epidermis is thick, it is necessary to mix with the water a small | beautiful, either way. $ ANSWERS TO CORRESPONDENTS, . reotype apparatus. We would scarcely advise any one to omigrate to Hayti. MINOR QUAM MINIMUS: The references in the first Volume of Cassell's R. SIMON His plan is excellent.-WM. JOXES should study the Lessons Classical Library are to Andrews and Stoddard's Latin Grammar, which is from the W. M. F. first.R. W. (Greenwich): Right.-J. BATES (Halifax) now in the press. is wrong about the formula to which he refers at p. 103, vol. iii., and we are Znta : As far as we can make out from the roughly-drawn MS. characters, right.-GEORGB DEWDNEY (Red-hill): His solution of the Wolf question the following are their equivalents:-os, T, ov, kas, ß, 7, 9, B, Š. They are was algebraic and elegant, but we have not room for it.--AN IGNORANT now only to be met with in old books. The German letter s may be written BOLTONIAN asks whether it be lawful for a lady to marry her father's half brother; 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 Pogla: We cannot undertake to correct exercises. the halves; this list is taken verbatim from an edition of the Bible, printed ZETA (Windsor): The vocabularies need only be learnt one after the other, and published by royal authority at Edinburgh, in the year that Queen as they are wanted, not altogether. Victoria was crowned (1838); W. R.C.: You are right in your supposition. “A TABLE OF KINDRED AND AFFINITY, wherein whosoever are related PHILOMATH : The word is intended for εστι. You mistranslate the other are forbidden in Scripture, and by our Laws, to marry together. sentence, which should be rendered thus: “Dishonour follows wickedness." A man may not marry his A woman may not marry her The article of the noun in the genitive may sometimes be omitted, but 1, Grandmother 1. Grandfather should generally be used when the English definite article occurs. The letter 2. Grandfather's wife 2. Grandmother's husband is never pronounced soft. Lessons in elocution will shortly 3. Wife's grandmother 3: Husband's grandfather appear. 4. Father's sister 4. Father's brother J. R. H.: The first Lesson in Chemistry is contained in Part XVI., No. 5. Mother's sister Mother's brother 7, p. 259, vol iii.--D. McK. (Row): Water can never rise higher than its 6. Father's brother's wife 6. Father's sister's husband level in any system of pipes, unless the forcing -pump be employed.--EIN 7. Mother's brother's wife 7. Mother's sister's husband JUSGER IRLANDER: It is impossible to tell which country has produced 8. Wife's father's sister 8, Husband's father's brotirer the most eminent men as compared with the population, unless you sit 9. Wife's mother's sister 9. Husband's mother's brother doggedly down, and with the census of a given period in the one hand, and a 10, Mother 10. Father list of the eminent men who hare fiourished within that period in the other, ll. Step-mother 11. Step-father compare the numbers of the former with the numbers of the latter, for Eng, 12. Wife's mother 12. Husband's father land, Scotland, and Ireland. Looking at the smallness of the population, and the great number of eminent men who have arisen in Scotland, we guess, 14. Wife's daughter 13. Daughter 13. Son 14. Husband's son but we only guess, that the result of the comparison would be in favour of 15, Son's wife 15, Daughter's husband that country. 16. Sister 16. Brother POWER-LOOM WEAVER : By all means study our Lessons in Geometry 17. Wife's sister 17. Husband's brother in the P. E.; they will make a man of you.-QUAESITOR ANXIUS (Jersey): 18. Brother's wife 18. Sister's husband Your friends are right about the expense of becoming a civil engineer. 19. Son's daughter 19. Son's son To give you an outline of the requisite studies would fill a volume. The 20. Daughter's daughter 20. Daughter's son best outline we have seen is contained in the programmes of the “Ecole 21. Son's son's wife 21. Son's daughter's husband Centrale des Arts et Manufactures," &c., at Paris : 1°. Programme des 22. Daughter's son's wife 22, Daughter's daughter's husband Connaissance exigée pour admission à l'Ecole Centrale ; 2°. Programme des 23. Wife's son's daughter 23. Husband's son's son Cours, première Année; Deuxième et Troisième Année, &c.-AMATEUR : 24. Wife's daughter's daughter 24. Husband's daughter's son Thanks for his sketch of a portable laboratory; the first chemists have 25. Brother's daughter 25. Brother's son liad only a few things collected on an old tea-tray.-GEORGE R. (Old 26. Sister's daughter 26. Sister's son Fish-street): See Dr. Beard's Lessons in English in the P. E. 27. Brother's son's wife 27. Brother's daughter's husband F.F. HENBEST (Fordingbridge): We shall be most happy to be favoured 38. Sister's son's wife 28. Sister's daughter's husband with a further explanation of his ideas on the motions of the earth and the 29. Wife's brother's daughter 29. Husband's brother's son moon, either by diagram or machinery; an annual rotatory of the earth on an 30. Wife's sister's daughter 30. Husband's sister's son axis perpendicular to the plane of its orbit, and a new explanation of the moon's monthly rotation, are certainly novelties worthy of our attention. PHILO (Nottingham): His corrections on the Key to Cassell's Arithmetic are all right; a few of the answers in the Key were taken from the LITERARY NOTICES. answers published by the original authors of the work, trusting that they were correct; but whether the errors he has discovered arose from mis FRENCH.. prints, or carelessness in the calculators, it is difficult now to say.-- CASSELL'S FRENCH DICTIONARY, in Numbers, 3d. each ; Parts, ls. each. W. WALKER (Southport): In the last edition of " Keith on the Globes," in The French-English division now ready, in stiff covers 4s., or strong cloth 1851, you will find the number of the constellations stated at 96, as taken 5s. The entire work will shortly be ready, price 8s. 6d. strongly bound. froin the Royal Astronomical Society's catalogue; and 9 more are added as CASSELL'S CODIPLETE MANUAL OF THE FRENCH LANGUAGE. By Progiren by foreign mathematicians. fessor De Lolme. Price 3s. in neat cloth. W. Jones (Stockport): The Lessons in Arithmetic are not completed; your writing is rather stiff, and your spelling is very bad.-R. BLOMELEY: 12s. in stiff covers, or 2s. 6d. neat cloth. Part II. will be ready in a few days CASSELL'S LESSONS IN FRENCH. Part I. By Professor Fasquelle. Price 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 CASSELL'S KEY TO THE ABOVE LESSONS, ls. in paper covers, or Is. 6d. di Mecanique Celeste;" there is a collection of tables called the “Requisite neat cloth. Tables" to be used with the “Nautical Almanac."-AN OLD SUBSCRIBER CASSELL'S SERIES OF LESSONS IN FRENCH, on an entirely novel and (Shrewsbury): We should be glad to please him by inserting his letter : but simple plan. Price 6d., or per post 7d. it is contrary to our rule to lend our pages to puff or praise any works of LATIN. which we do not know the value. w. W. V.: To the exercise appended to Prop. II., Book IV. of Cas. | drews and Stoddard. Revised and corrected, price 3s. 6d., will CASSELL'S GRAMMAR OF THE LATIN LANGUAGE. By Professors An shortly be sell's Euclid, add the words "only in the case of the equilateral triangle.”. ready. JOHN DAVIES (Pontypool) should get a copy of Cassell's “ Emigrant's CASSELL'S LATIN DICTIONARY. By J. R. Beard, D.D., will be issued in Almanac," price 6d. Weekly Nurnbers at 3d., immediately on the completion of the French S. Gent (Sutton-in-Ashfield): His method of calculating discount is no Dictionary. The entire price of the volume, bound, will be 85. 6d. 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): CASSELL'S LESSONS IN LATIN. Price 28. 60. paper covers, or 33. neat cloth. 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 CASSELL'S KEY, TO THE ABOVE LESSONS. Price 18. in paper covers, or study of two languages together is very likely to lead to confusion; we would ls. 6d, neat cloth. therefore recommend that of a language and a science together.-W. F. P. CASSELL'S FIRST LESSONS IN LATIN. By Drs. Andrews and Stoddard. (London): “A farmer han £100 to buy oxen, sheep, and geese, so that he Price 1s. paper covers, or Is. 6d. neat cloth. should have 100 animals in all, the oxen being £5 a-piece, the slieep £1 GERMAN. al-piece, and the geese ls. a-piece. Ilow many of each did he buy? Here, let x, y, and % denote the numbers of each sort of animal; then by the CASSELL'S GERMAN DICTIONARY is now issuing in Numbers, at 3d. each question we have these two equaticns : -(1.) x+y+z=100, and (2) Monthly Parts, 1s. each. 100x+20y+x=2000; subtracting (1.) from (2.) we have 99%+-19y=1900, CASSELL'S LESSONS IN GERMAN, price 28. in stiff covers, or 28. 60: 1900-9930 cloth. and therefore ca 100-5.0 + ; now, as there can be no 19 19 42 fractions, 79 must be a whole number; in order that this may be the case, Fables. vol. I., price 1s. 6d. in neat cloth, is now rcady. Vol. II. is in CASSELL'S CLASSICAL LIBRARY, containing a Latin Reader, with Easy 4x must be divisible by 19; and therefore x=19; whence y=1, and x=80* course of issue in Weekly Numbers, at 2d. cach, consisting of Latin The answer is, therefore, 19 oxen, I sheep, and 80 geese. The other Exercises intended to ground the student in the Syntax of the Language. question is useless, and so plain that it needs no explanation. 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 J. E. (Kidderminster): Thanks for his corrections; if any sections seem . to he omitted, it can only be by mistakes in the numbering. - Gautier carefully revised and corrected. CASSP.LL'S EUCLID.-THE ELEMENTS OF GEOMETRY. Containing the (Bristol): “ Uócle Tom's Cabia” in French, and the“ Key” to the same, may be obtained at our office.-P. H.: We would recommend the study of French First Six, and the Eleventh and Twelfth Books of Euc:id. Edited by Professor and Latin together to those who have time at their disposal.- A Reader Wallace, A.M., price ls. in stiff covers, or ls. 6d. nent cloth.-KEY. 31. in Ulvertone, whose signature we cannot read, is referred to Messrs. CASSELL'S ELEMENTS OF ALGEBRA (uniform yrith Cansell's EUCLID), Horne, Thornthwaite and Co., 123, Newgate-strect, for the price of daguer- price ls, in stiff coyers, or ls. 6d, neat cloth, INSTRUMENTAL ARITHMETIC.-No. III. quadrant, the angle Bo A' the second quadrant, the angle a'o B the third quadrant, and the angle B'oa the fourth quadrant THE PLANE SCALE AND PROTRACTOR. the initial position of the revolving line being always 0 A. (Continued from page 13.) Fig. 1. 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 definition is greatly altered and extended. Angular mag. nitude 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 om be its first terminal Now, if the terminal position of o A be in the first quadrant, position, A OM is an angle in what is called the first quadrant, is called the first quadrant, as om, the angle A ox is said to be less than and is less than a right angle. the terminal position of 0 A lie in the second quadrant, as om', In order to explain the different quadrants, it will be suffi- the angle A 0 m' is said to be less than two right angles, and Fig. sent to remark, that by the 2nd corollary to Prom. SIII., I greater than one right angle; if the terminal position of 0 Aassell's Euclid, all the angles about a point are equal to four lie in the third quadrant, as on, the angle is said to be ght angles ; and if a d' and BB' be two straight lines drawn greater than two right angles, and less than three righư at right angles to each other, they form four right angles, viz., angles; and if the termina position of o A lie in the fourth A O B, BOA', A OB', and B'O A. Here, each of these angles is quadrant, as o N, the angle is said to be greater than three one-fourth, or a quadrant, of the entire angular space about the right angles, and less than four right angles. Moreover, if pomiv; and by convention, the angle A o B is called the first the terminal position of oa be such that it has passed once SO on. over o A in its revolution, and reached it a second time, then the angular point o cut off a certain part om of the revolving it lies in the fifth quadrant, and the angle is said to be greater straight line which generates the angle A O M, and from the than four right angles, and less than five right angles ; and point m draw MP perpendicular to the straight line o A, its initial position, we shail then form what is called the ElemenAngles are in practice measured by the arcs of circles inter- tary Triangle omp. In this triangle, which by construction cepted between the initial and terminal positions of the is right-angled, the ratio of the perpendicular MP to the hyporevolving straight line. The circumference of every circle tenuse om, is called the sine of the angle AOM; the ratio of the is by convention divided into 360 equal parts called 'degrees, perpendicular MP to the base o p is called the TANGENT of the and marked o; for minuter parts, the degree is divided angle A OM; and the ratio of the hypotenuse on to the base into 60 equal parts, called minutes ; for second minuter op is called the SECANT of the angle AOM. Again, the ratio of parts, the minutė is divided into 60 equal parts called the base o p to the hypotenuse om, is called the COSINE of the Seconds; this is called the sexagesimal (from Lat. sexa- angle A Om; the ratio of the base o p to the perpendicular MP gesimus, the sixtieth) division of the circle. In France they is called the COTAXGENT of the angle A om; and the ratio of the adopt the centesimal division of the circle, but it has its dis- hypotenuse o M to the perpendicular mp is called the COSECANT advantages. In the sexagesimal division, it is plain that a of the angle AOM. The latter three ratios might have been right angle is measured by a quadrant of the circle, hence it defined as the reciprocals of the former three ratios ; thus, the is said to contain 90°; two right angles are measured by two cosine is the reciprocal of the secant; the cotangent is the reciprocal quadrants of a circle, or by a semicircle, and are said to con- of the tangent; and the cosecant is the reciprocal of the sine. tain 180°; three right angles are measured by three quadrants Referring to fig. 1 again, if we take the angle A O M' in the of a circle, and are said to contain 270o; and four right angles second quadrant, and from the angular point o cut off a certain are measured by a complete circle, and are said to contain 360°. part o of the revolving straight line which generates the Moreover, five right angles are measured by five quadrants, angle A O M', and from the point m' draw M' P' perpendicular to and are said to contain 4500 ; and so on. the initial straight line o A, produced to s', we shall then form The instrument used for measuring angular space, or angles the Elementary Triangle o n' P' for the ratios belonging to the in general, is called the Protractor, as shewn in fig. 2, and angle A OX'. Here, as in the preceding case, the ratio of the consists of a brass semicircle graduated (that is, marked with pendicular M' P' to the hypotenuse o m' is the side of the angle degrees) from 0° to 180° either way, so that every arc has its 40M'; the ratio of the perpendicular m' rto the base o p’ is supplement marked along side of it, there being two rows of the TANGENT of the angle A OM'; and the ratio of the hypoteplumbers, one from right to left, and one from left to right; nuse o si' to the base o p' is the secant of the angle A OM'. that is, as in the figure, 0° begins at A, and 10°, 20°, 30°, &c., Also, the ratio of the base o P' to the hypotenuse o m' is the are parked on the outer edge of the instrument, terminating Cosine of the angle A o M'; the ratio of the base o p' to the here to B, which is marked 180°; and again 00 begins at B, and 100, perpendicular n' p' is the COTANGENT of the angle AO M'; and 20°, 30°, &c., are marked on the inner edge of the instrument, the ratio of the hypotenuse o m' to the perpendicular m' ' is terminating at A, which is marked 180°. the COSECANT of the angle A O M'. The use of this instrument is to protract (Lat. protraho, to The trigonometrical ratios belonging to the angles A 0 M and traw out), that is, to lay down an angle of any given number of 40.m' above explained, are usually exhibited in the following degrees; it is also employed to measure any given angle, that abridged forms :is, to ascertain the number of degrees which a given angle First Quadrant. contains. Besides the semicircular form of the Protractor, there is = sine 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 tangent Juler, 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 Semicir. cular Protractor are transferred to the Scale Protractor, which of the angle A OM. is marked and numbered in the same manner as the former, = cosine by placing a ruler or straight edge on the centre c, and on the several divisions of the semicircumference A D B in succession ; = cotangent then marking on EF, the edge of the scale, the intersections of the straight edge with that edge by portions of the straight lines 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 Second Quadrant. the figure, is the blank edge of the Scale Protractor. Thus you see M'p' that in the latter, three of its edges are occupied with the trans sine ferred graduations of the semicircle, and the fourth edge is blank, but contains the centre of the semicircle c, which is at M'r' tangent an equal distance from the extremities of the scale. Although we have combined these instruments in the figure, to show OM' their construction, the student is not therefore to suppose that OP' there is any such combined instrument in use; either will of the angle A OM. serve the same purpose, but one of them is enough in a case = cosine of instruments ; the Semicircular Protractor is perhaps more OM' handy in practice, but the Scale Protractor is more portable, cotangent and in some cases is more useful than the former. The graduated edges of both these instruments should be bevelled ом" almost to sharpness, in order to admit of the easier pointing M'p' off of the divisions or degrees of the limb, the name applied to the graduated edge. In some cases, Protractors are made the usual contractions for these and some other ratios are completely circular; and for many purposes they are highly useful, especially for laying down plans in surveying. given in the following table :In Trigonometry, which originally meant the measuring of tri Names of Ratios, Contractions. angles, there are certain straight lines drawn in and about an sine sin. angle, whose ratios to one another are called Trigonometrical tangent Functions, or simply Trigonometrical Ratios. Referring to fig. 1, if we take the angle A om in the first quadrant, and from MP OM MP OP OM secant OP OP OM OP MP OM cosecant MP OM OP secant OP' OP' M'P cosecant cosine tan. secant sec, COS. |