the 11 wo Examinations. By A. H. Dick, M.A., Lecturer on light upon the much disputed piece of prophecy in that, in each of these positions, one of its ends will be on AB, or BC, and the other on the gow. Op. 232.) London: Longman and Co. The A. B. C. Simplified, and Reading Made extensions of those lines through B. Wherever 1863.-The present work is introduced by a Pre- Easy to the capacity of Little Children. By G. The point M may be, within the angle ABC, the fatory Notice from the pen of Mr. J. S. Laurie, G. Vasey. Montreal : Lovell; Toronto : Miller. I latter two positions of the line will always be one of Her Majesty's Inspectors of Schools. After We cannot see that the author has in these pages, possible, but not the former. In fact, it is easy stating that “110 complete special Compendiuin by his system of “Rudimentary Combinations, to assign an area-locus within which M must fall, of Mathematical Geography exists in the English established a royal road to reading. The work, berween the lines AB, BC, in order that it may tongue,” Mr. Laurie informs us that, having sought however, is well adapted for beginners, as its cost be possible to draw through it a line which shall for the means of supplying the deficiency in Ger- is very small, and its type clear. exactly satisfy the conditions of the Problem, that many, he met with a clear and concise Exposition is, which shall have the purt intercepted by AB, of the subject in Brettner's “Mathematische Geo BC, and not their continuations, equal to the given graphie," the translation of which, undertaken at MATHEMATICAL QUESTIONS AND length (a). If ABC be a right angle, the area. his request by Mr. Dick, forms the basis of the SOLUTIONS. locus will be a quadrant of a given tour-branched present publication. Mr. Dick has, however, hypocycloid, and when ABC is oblique, it will be made such additions to the German original, 1264 (Proposed by Geometricus, Brussels.) a sector of the area of a curve of similar form ; rendered necessary by the omission of some im im: On donne un angle ABC, un point M et une the bounding curve, in both cases, being the .On donne un angle ABC, un point M et portant branches of the subject, and want of ful. I longueur a, et on demande de faire passer par le envelope of a line, of length a, which moves with ness of illustration in others. that it may be con point M une droite telle que la partie Y inter-lits and sidered as substantially a new work. As it is, it ceplee entre les cotes de l'angle soit egale ål indefinitely extended lines, and a point in their fills an acknowledged gap in our educational lite. droite donnée a. plane, be given, it is possible to draw through the rature. Whether viewed as a text-book, or as a SOLUTION BY W.J. MILLER, B.A., MATIEMA point always two, sometimes four, but never work of reference for the Teacher's own guidance more ihan four, straight lines such that the part TICAL MASTER, HUDDERSFIELD COLLEGE. in giving instruction on the various subjects of intercepted between the fixed lines shall be of which it treats, it will be found equally useful, given length, Note on Mr. MILLER's Solution of Question 1264, by Mr. S. Bills. elucidated by an analysis and Explanation of This Prolilem may be constructed by means of the Fables of Nathan Covington Brooks, A.M. a circle and hyperbola, as the following analysis London : Sampson Low. New York : Barnes and will show. Referring to the diagram in Mr. Burr.- Among the numerous editions of Ovid this Miller's solution (of which a proof slip has been handsome and scholar-like treatise merits a word kindly forwarded to me), put XY = a, BP = b, of praise. Mr. Brooks is evidently imbued with MP = C, cos ABC=1, BX=%, and BY=y. no ordinary love of his author, and has collected, From the similar As BXY, PXM, we have from a great variety of sources an amount of r: y= -6:c, or, xy-by=cx ......(1). illustration, in the shape of critical annotation, We have also 2? + y: - 2xy=a:..........(2). parallel passages from ancient and modern authors, and other aids to the study of the Metamorphoses, X |(1) 22 + (2) gives ro + y2 – 21 (cx + by)=a*.. (3). such as, we believe, are not to be met with in any Analueis - Suppose XMY to be the required And (1), (3) may be put under the forms other edition of this favourite school-book. Many line: construct the parallelograms BXOY, BPML, (x-1) (y-C)=bC.................. (4); of the fables, whose resemblance to the biblical and let the diagonals (XY, BO) of BXOY meet (x-ac)2 + (y-26)=a? +1? (19 + c)........(5). records are so obvious, as to suggest the idea of in D. Then XD = DY, BO = 2BD; and, byl their being corrupt traditions of Scriptural truths, similar triangles, we SD; and, uy Now, x and y may be determined (geometrically) | similar triangles, we have have been traced back to their origin, and illus from the intersection of (4) and (5); but, referring trated by parallel passages from the Scriptures. PX: PM = LM;LY, the system to rectangular coordinates, (5) will reA lexicon, giving the meaning and derivation of or, MN: PM = PB : NO; present a circle, the coordinates of whose centre all the words in the text, together with a very full ir parallelogram OM = BM, (Euc. vi. 14). are (ac, ab) and radius= /(a? +226 +1°c); and scanning table, add greatly to the value of the Hence, M being a given point, the area of the |(4) will represent an equiluteral hyperbola, having work. lits asymptotes parallel to the axes of reference, parallelogram OM is constant; therefore, when The Six Standards of Arithmetic: Standard 11. XY passes through M, the locus of () (the corner, the coordinates of the centre being (, c), and By Walter M'Leod, F.R.G.S. (pp. 92.) Lon- remote from B, of the parallelogram on XY as semiaxis = 7(260). don: Longman and Co. 1863.-- This is a con. diagonal) is the given hyperbola OHK, along the tinuation of the Series of Arithmetical Standards, asymptotes LMN, PMQ. now in course of preparation by Mr. M‘Leod, tol Again, if a straight line XY, of constant length, 1321 (Proposed by W. J. Miller, B.A.. ineet the requirements of the New Code. It takes moves with its ends on two fixed straight lines Mathematical Master, Huddersfield College.) the pupil to the end of subtraction. Its special | AB, BC, the locus of its middle point (D) is, by | If two marbles are thrown at random on the floor feature is the way in which those formidable a well-known property, an ellipse of which B is of a square room, what is the chance that they stumbling-blocks to beginners, Notation and Nu-l the centre, and the major semi-axis is along the will rest at a distance apart less than one-fourth meration, are dealt with. Instead of bringing the bisector of the angle ABC. But BO=2BD; of the breadth of the room? little learner at once face to face with bewildering therefore, when XY is of constant length, the series of billions and trillions, which for the most locus of O is a given ellipse. When, therefore, General Solution by W. S. B. WOOLHOUSE, part excite no definite idea in a child's mind what. | XY both passes through a fixed point (M) and is! F.R.A.S., F.S.S., &c., Alwyne Lodge, London, ever, Mr. M.Leod proceeds on the principle of of given length (a), as in the Question, the posi. divide et impera-first a little easy Numeration ;tions of o, and consequently of XY, will be Take the more general then a little addition, whereby the kuowledge determined by the intersections of the ellipse problem for a rectangle gained is practically applied; then a little more and hyperbola. Hence we have the following ABCD whose sides are numeration, followed by some more addition or Construction. Make BA = BC = a; on the DL CAB=a, AD=b; and let subtraction, until the simple rules are gone through, bisector of the angle ABC take BS, such that SZ K' the limiting distance be. pari passu with the more difficult parts of Notation. ( BS) = a, Z being the intersection of this per. tween the marbles (P, Q) This is an obvious improvement on the ordinary pendicular with BA produced ; also on the bi. F =r, and the coordinates system, which the intelligent teacher will not fail sector of the angle NMQ take MH, such that the Q.TTTT Sof P, AH=4, PH=y. to note. rhombus on MH as diagonal may be equal to] A Ħon En BRF, London: pub. / semi-axis, draw the ellipse ASC through the drawn with P as a centre, the other marble Q, lished for the Author by H. J. Trender.-The points A, C; and with H as vertex, HM as major according to the proposed conditions, must lie in author of this little work has sought to represent semi-axis, and MN, MQ as asymptotes, draw the that portion of the rectangle which falls within pictorially some of those mystic animals in the hyperbola OHK, intersecting the ellipse ASC imobia llipse ASC in this circle. But if lines EF, GH be drawn Book of Daniel which have given rise to so much 10, K; through 0, K draw OX, KX parallel to! to through P parallel to the sides, and dividing this discussion, not always very profitable. It is to be BC; and through X, X' draw XMY, XMY';} i particular surface into four parts, it will be evi. recollected that while such writers as Lord Carlisle, either of these two is the line required. dent that, when P takes all positions on the and Dr. Cumming unhesitatingly assume the au. The proof is evident from the analysis and rectangle, each of these parts will pass through thenticity of the book of Daniel. many divines, / construction. precisely the same values, so that a calculation including the late Dr. Arvold, bave entertained Scholium. It is evident that the other branch with respect to one of them will be sufficient. If grave doubts about it. The work before us is, of the hyperbola will intersect the ellipse in two therefore s denote the surface of that part of the perhaps, harmless enough; nevertheless, we do points which will determine two other positions circle which falls on the rectangle PA, the numnot think that it is calculated to throw any new of the required line; but it is to be observed ber of positions conformable to the question will dSxbX', and But die al limi But tained within the circle is '; . 49=y'; andsince dy When r<a, p= (-*** ) When r>a, P be represented by 4 S/dxdys, the total number of possible positions being ab x ab = a2b2. The For y=0....z', S=xy and sdy S=1&x”. sought probability is therefore For y=x'....b, dans =y", and saxSxo=xSqb+jx®3 (between limits) P J dxdys. Sdys=yS + $y'3 (between limits) =aSxo - U (66')-313 + ja’3; There are three distinct cases which it will be =63x6—x(x:x°) – 3x3 + 38°3. 1.- SSdxdyS = abSab+} (a”?b + ab'3) + . * – 464 convenient to investigate separately. Case I.-When the distance r, within which the Therefore for the total limits –6961-9644a22 + 46+2p?......(2). marbles are required to lie, is less than b. S dyS=6S.[6–1xx'? – 3x3 + {23 Adding (1) and (2), and putting go - 62 for 7", With radius r and centre A draw the quadrantal we get for the whole rectangle ABCD arc db, and through d, 6 draw dh', ab parallel to =65x6—*rx +4x3 + 363. the sides. Also to avoid the use of surds, when I And, integrating for x='....t, SA dädys = abSab+} (a3b+ab3) + x1 (at +64) x and y are either of thein less than r, let x', y'! - * (a2 + 14) p2 – , denote complementary coordinates such that x2 + 1 dxdyS=bf dxSxb-36'4 +46°3+ 4622 pt. where m'? =y2 + y = r. First consider the positions of P on the rect dszb=&', / xdsxb=-*'3 ; and Sof=*+} (aa' + b1')—(cos-to+cos-) angle Aa. dx When y passes from 0 to x', $ = xy and also {(a'3b + ab3)=} (d'b + ab')2 s dxSx6=xSxo+fx"3 (between limits) ab (aa'+bb'); S dys = {xx”. =rSpo—V (66')- 363 ; When y= x'......, the portion of AH con 1:://dxdyS = brs, -6°42-564 – 36'4 + 38°3r + 8,- .............(2). ydy=-ydy, fyds=-**; Lastly, when Pis on the rectangle kc, :: Sdys=yS-syds :: dyS=yS + {y'3 (y=0......) =yS+ {y'3 (between limits) =65, +383 – 17.3; =rS...—." (02') – gião. :: sdxdyS=(a-r) (65,. + }b'3— 3)...... (3) When y=r....b, S is constant, and Adding (1), (2), (3), and substituting p2-62 for in which o denotes the same form of function as Therefore, collecting these three values, we get, dedys = abSabt ab'ra-ab' + 464-1692 in Case II. for the total limit y=0....b, When the proposed figure is a square, as in the - far, I question, then a=b, and Case II, is not needed. Gdy S = 65.<r – $xx"2–3x3 where Sab=1#r2+361'- rocoso; =652r- ror + fr3. Integrating again for x=0....t, ffdædy8=b/dxSxr-o r. But "Sev=x' and, since xde=-x'da', AxdS.cp=-37" ; where 1341 (Proposed by W. J. Miller, B.A., Mathematical Master, Huddersfield College.) – :: fdxSxy=xSxr + fx"3 (between limits) A piece of slender uniform wire, of indefinite =>Spy – fr3; length, is bent into the form of the curve (x + 4a)3=27a (x + y). - 2 cos-12 ::SS dxdyS=brs, - 463-nin .......... (1). 41 161 If A,, A,, A, be the attractions, on a particle at Secondly, consider the positions of P on the rect-l is a small function. of the 1.2 287 the origin, of the whole curve, of a loop of it, and 1.3 0489 angle 6C : fifth order with respect to *0893 of the part which exerts the greatest attraction towards the vertex, prove that When y=0....",">=y' and, as before , which admits of con A, :A, : A, = 8:6/3: 19. Also find an arc of the curve which would alone s dyS=yS + {y'3 (between limits) venient tabulation. The 844 exert the same attraction as the whole curve. marginal table is a brief =rs, - 3r3. specimen, and one a little When y=r....b, S=S, is constant, and more minute and extended SOLUTION BY THE PROPUSER. would give the values acf dys=(6-) S, curately by inspection. If The attractions are most easily obtained from Hence, for the total limits y=0....be we put b=r cos 0, the function will take the fol. the corresponding polar equation of the curve, which is pora sec3 10, :: ffdxdyS=(a—r) (6S, -) ............(2. from which a more extended table may be con o being estimated from the negatire branch of the axis of ix. For the whole rectangle ABCD add (1) and / veniently calculated. The curve is, in fact, the envelope of a perpen. (2); the result is Case III.- When the distance r is greater than dicular, at its extremity, to a focal radi Sfaxdys = abs, - } (a+b) ** + firm, a and less than v(a? + 7%). parabola whose parameter is 4a, and its form and First, supposing P to be on the rectangle Am', le rectangle Am', properties are very fully investigated in my Solu, where Sforo; we have S=xy, aud tion of Question 1283, in the “Educational Times" for December. ffdxdys -40°62.... ..........(1). Now the resultant attraction on a particle at the Secondly, let P be on the rectangle n'C, and the origin (i.e. the focus) of a part of the curve, sy Case II. When the distance r is greater than integral with respect to y will be the same as in metrically situated on both sides of the axis, Will b and less than a. the rectangle nh of Case II. ; be along the axis; hence, if A be the attraction With radius r describe the quadrantal arc emk, . ,,s=os,-trt113 11.212. lowards the vertex, and x a constant, we shall and draw mn, kh parallel to DA. First let P be on the rectagle Am; then S=xy, Jand, integrating for x=b'.... a, A= / xp? cos 0 ds=/xa-l cos } cos de .: S drdys=*x*y=*6*6** ............ (1). dxdys=b | dxSxo + 40+ - 88'4 + fab'3 – faoy = (15 sin 10-25 sing je + 12 sino FO); Secondly, let P be on the rectangle nh: + 46°37. dc 10 *0000 3 10a 9x M.A.; Mr. W.RENWICK ; D. Sampson: ::A, -(A, from 6=0 to 0= 1344 (Proposed by Rev. W. Mason, Nor- Three persons, A, B, and C, draw out, in sucmanton, Yorkshire.)-Å Surveyor was brought cession, each of them a coin. What is the value to one of the Angles of a Triangular Plain, ibe of the expectation of each ? Solution by * * *; and Mr. F. SINCLAIN. that the other Angles were inaccessible, but that the distances from the angle where be stood to More generally ; let there be n coins, and let Ag =(A, from 0=0 to e =*) 204 the Vertices of two Equilateral Triangles described their values be a,,a, ..............6; on the opposite side were known. Then, said he, ..A, : A,: Ag=8:673: 19. no survey will be required: the Area is equal to also let s=a, +ag + ........ta Again, let a be the angular extent, on each side the product of the sum and difference of these of the axis of an arc wbose attraction would be distances, multiplied by the decimal 1443376. Then A's expectation is evidently.; equal to that of the whole curve; then, putting 15 sin 10-25 sin3 10 + 12 sinó 10=F (), Solution by J. VIPOND, B.A., Huddersfield Bie College; Mr. S. Bills ; Mr. W. Hopps; Mr. we shall have to find a from the equation W. H. LEVY; Rev. W. MASON; R. J. Nelson, n - 1 For shortness' sake, put sin =1; then , will be R. TUCKER, M.A.; J. R. W., Jesus College, Cambridge ; and *** 'C's will be given by the equation 2 1226 - 25x3 + 152–2=0, ($-a,-&2 or, dividing by (2-1), Let A be the area of the "(n-1)! n-2 1-2 1213 +2412 + 111-2-0. triangle ABC, S the sum, - n(n-1) 8-2 (n-1)s _s and D the difference of the This cubic equation has only one real root, and its lines (AP, AQ) drawn from = (x-1) (1 – 2) = numerical value may be most readily found by the accessible angle (A) to Therefore, the à priori expectation of each is the Horner's method. the vertices (P, Q) of the It may however, by Cardan's method, be ex two equilateral triangles same and=-. pressed in the form constructed on the base 1= [V (20+5/11) + (29–5 /11)-4]. BC. In the particular case proposed in the Question, Both these methods give Join PQ, cutting BC in we have n=12, s=21. 12s. 2d. 1=sin fax.1376397=sin 7° 54' 40.64"; 0; and draw AR perpen. Therefore the expectation of each is 4s. 4fd. dicular to PQ. Then we ..q=23° 44' 2". have Hence an arc of the curve, extending from the 1349 (Proposed by Mr. D. Sampson.)vertex to an angular distance of 23° 44' 2" on each AQ?-AP2=RQ:-RP2=4 OR. OP The trilinear equation of the circle which passes side of the axis, would exert the same attraction =40R.OC_3=4 A 3; through the three centres of the escribed circles as the whole curve on a particle at the origin. of any triangle of reference is .A =S x D x (v3) aa? + bB2 + ca? + (a+b+c) (By+ya + ab) = 0. 1342 (Proposed by Dr. Rutherford, F.R.A.S., =Sx D x •1443376. Royal Military Academy, Woolwich.) - Four Cor. It may be easily shown that the triangle Solution by ***; Mr. D. SAMPSON; J. R. W., equal cylindrical rods are placed symmetrically ABC is one-third of the difference between the Jesus College, Cambridge ; and the Rev. R. H. round a fifth one of an equal radius, and three equilateral triangles constructed on the lines AP,I WRIGHT, M.A. equal elastic bands surround the bundle, one AO. The equation of any circle may be written being in the middle, and the other two equidistant from it. A weight of 1 lb. would just stretch aby + bya + caß + (aa + bB + cy) (la + mß + ny)=0. each band to twice its natural length, which is If the centres of the circles escribed to the tri1345 1345 exactly equal to the circumference of each rod, (Proposed by W.O. Phillips, M.A.)- angle of reference lie on this circle, its equation and the coefficient of friction is **; what force E, F are any two points in the opposite sides in must be satisfied by each of the three systems would be required to draw out the centre rod, the AD, BC of a parallelogram, and P, Q the inter B=y=-a, y=a=-B, a=B=-y; sections of the lines AF, BE, and DF, CE. axes of the rods being vertical? Prove that the line passing through P, Q, bisects ii-l+m+n=l-m+n=l+m-n=1; Solution by J. R. W., Jesus College, Cambridge; the parallelogram. i.l=m=nel. Hence the equation to this circle is M.A.; Mr. J. CONWILL; Mr. D. SAMPSON; I aßy + bya + caß + (aa + bB + cy) (a +8+4)=0, and GEOMETRICUS, or, aa? +682 + cya + (a + b + c) (By+ya+aß)=0. A plane containing one of the strings; and let the figure represent half Let G, H be the hel Mr. Wright adds the following sir. Wrigot adds the follow this section. Let AE, BF, BG be points at which the Corollary.--The equations to the three circles drawn perpendicular to the di. line PO meets AB, which pass through the inscribed centre and each rection of the string. Suppose CD; draw EI parallel two of the escribed centres are the bundle placed on a horizontal to CD, and join AI, 6B2 + cya-aa? +(6+c-a) (By-ya-aB)=0 plane and the centre rod pulled DI, FI. Then, by cy? + aaa-b82 +(c + a-6) (ya-a3-BY)=0 vertically upwards, then if P be the force required, similar triangles, we 3R the pressure of each rod on the centre one, W have aa? +682-cy* + (a +b-c) (aß-By-ya)=0. the weight of the latter, we shall bave PG : PI=PB : PE=PF:PA; P=W+12 Rx = W+2=R. :: A APG = FPI (Euc. vi. 15); NEW QUESTIONS. A AGI=AFI ; and similarly A DHI=DFI.1 1361 (Proposed by W. S. B. Woolhouse, And if r be the radius of a rod, then Hence, fig. AGHD(= A AID+ AGI+ DHI = F.R.A.S., &c.)- If three marbles are thrown at EF=2r</2, FG=#r; AJD + AFI + DFI) = AAFD = } parallelogram random on the floor of a rectangular room, what :: stretched length of a string=267 +8r 2. ABCD. ' is the chance that the triangle which unites them Now if I be the natural length of an elastic! That is, GH bisects the parallelogram. will be acute-angled ? string, L its length when stretched by a tension t, Note.--As every line bisecting a parallelogram 1362 (Proposed by Mr. W. K. Clifford, . and e the modulus of elasticity, we bave passes through the intersection of the diagonals, Russell Square, London.)-For every point A on it is obvious that this Question is a particular a rectangular hyperbola, there exists a straight case of Prop. 139, bk, vii., of " Pappus's Mathe. (line BC, passing through the centre, such that it matical Collections.” But, unlike that proposi- through any other point D on the curve, lines be hence, since I lb. would stretch a string to twice tion, it admits of a geometrical solution inde-drawn parallel to the asymptotes, cutting BC in its natural length, e=1; also, if T be the tension | pendent of the doctrine of Harmonicals, as ap- B, C, the intercept BC subtends a right angle of the string when surrounding the bundle, pears from what precedes. See the solution of at A.' | Question 1139, in the Educational Times for 1363 (Proposed by Mr. W. H. Levy.)-The internal bisector of the angle B of a triangle ABC Again, for the equilibrium of an external rod, meets the side AC in F, and the circumscribed 1348 (Proposed by Mr. S. Bills.) - One circle in H; from F, H, perpendiculars are drawn sovereign, two-half-sovereigns, one crown, two to the side BC, meeting it in 1, Q; and FI is pro half.crowns, one shilling, two sixpences, one duced to meet a semicircle on BQ in P; prove Hence, P=W + 2xR=(W +16) Ibs. I penny, and two half-pennies, are put into a bag. I that AC=2PQ. L=1(1+); 2«r + 85 /2=2ar (1 + T), :: T=-12. R=2T cos 45o= 1364 (Proposed by Mr. Stephen Watson.)- It was resolved that the Book of Genesis and 12mo, cloth, price 28. 6d., Let O be ibe centre of an ellipse, OC a radius the Guspel of St. Matthew should be the portions CHERVILLE'S FIRST STEPS TO vector, CD a chord perpendicular to OC, CP a of Scripture for the Pupils' Examinations in the U FRENCH. Indispensable to, and in harmony tangent at C, and OP a line bisecting CD and year 1864 : and that the next examination for the with, all French Grammars, being a collection of Prómeeting CP in P. Required the equation and College Diplomas should commence on Tuesday, shewing a parallel between the Pronunciation. Ett gressive Familiar Conversations in French and English. area of the locus of P. June 23rd. mology, Accidence, and Idioms of the parts of speech in 1365 (Proposed by Mr. S. Bills.)-Find two Thanks were voted to W. McLeod, Esq., and both Languages, with Grammatical Observations. to Mr. Gordon, of Edinburgli, for their donations *** Heads of Schools, &c., forwarding their cards to numbers such that both their sum and difference shall be a square, also the sum of their squares of books to the College Library. | Mr. W. TEGG, a copy will be sent gratis, in the hope that they will adopt the work if approved. shall be a cube, and the sum of their cubes a Memh The following ladies and gentlemen were elected London : WILLIAM TEGG, Pancras Lane, Cheapside. Members of the College :square, Mr. James Belcher, Darlaston. BLAND'S LATIN HEXAMETERS. 1366 (Proposed by Matthew Collins, B. A., Mr. Richard Bulmer, Grammar School, Whit DILEMENTS OF LATIN HEXAMEProfessor of Mathematics, 21, Eden Quay, Dublin.) | church. V TERS AND PENTAMETERS. By the Rer. R. --If a, b be the major and minor semi-axes of an Miss E. Carey, London Place, Ilack ney. BLAND. New Edition, corrected and improved by the ellipse, and R that radius of curvalure which is, Miss M. A. Cole, London Place, Hackney. Rev. G. C. ROWDEX, D.C.L. 12mo, 3s. cloth. both in magnitude and position, a chord of the Mr. G. W. Davies, Cliffe House, Lewes. A KEY to the above, adapted to this edition, 12mo, Mr. John Fieldhouse, Altrincham. 5s. cloth. ellipse, prove that, 4 (Rab)ī is an harmonic mean Mr. Charles Fisher, Slowbruy Villa, Sand Hutbecween ao and 69, and that the inclination of R to London: SIMPKIN, MARSHALL, and Co. ton. the minor axis is į cos. Mr. T. M. Jacombs, West View House, Hamp ADAPTED TO THE 'REVISED CODE: May now be had, in 18mo, 9d. sewed, MHE SIX STANDARDS OF ARITH1367 Mr. J. A. Jarman, University School, Notting. (Proposed by R. Tucker, M.A., Port-| 1 METIC, STANDARD II. containing a Graduated ham. arlington.)-0, 0, 0, are the centres of the Course of Exercises in Numeration, Addition, and su Mr. B. E. Pearson, Brackley. traction; with Methods of Solvinz the Questions. By escribed circles of the triangle ABC. Ifr ,?,,. Mr. James Royds, Rusholme, Manchester. WALTER M'LEOD, F.R.G.S., M.C.P., Royal Military Asylum, Chelsea. and R, R, B be the radii of the circles in. Jr. William Watson, B.A., University College School. STANDARD I. comprising 650 Questions in Mental and Slate Arithmetic, &c., price 9d. scribed in and described about the triangles London: LONGMAN, GREEN, and Co., 14, Ludgate Hill. 0,BC, 0,CA, 0, AB; then THE Pneumatic Dispatch Company having laid THE REVISED CODE. the North-western post-office in Eversholt-street, Preparing for publication, in Six PARTS, fcap. 8vo, o which Parts I. and II. will be ready in a few days, 1368 (Proposed by the Rev. R. H. Wright, inst, by the Postmaster-General, Lord Stanley of THE GRADE LESSON BOOKS, in ht; Alderley, Sir Rowland Hill of the Post Office. 1 Six Standards; each embracing Reading, Spelling, M.A., Trinity College, Cambridge, and Head and several and several other gentlemen. The working of | Writing. Arithmetic, and Exercises for Dictation. Ex Vlaster pecially adapted to meet the requirements of the Revised ol, Ashford, Kent.) this novel mode of transit was satisfactory to those Code. By E. T. STEVENS, Associate or King's College, - Find the equations, in trilinear coordinates, of present the letter.bgos brought up by the North London; and CHARLES HOLE, Head Master, la the circles whose diameters are (i) the perpen. Western Railway, which occupy ten minutes in! borough Collegiate School, Brixton, late Master of St. diculars drawn from the angles of a triangle to the carriage to the Eversholt-street post-office Thomas's Collegiate School, Colombo, Ceylon. the opposite sides, (ii) the lines bisecting the when conveyed in the usual way, having been London: LONGMAN, GREEN, and Co., 14, Ludzate Hil angles and terminated by the opposite sides, and blown through the company's tube in about a Price 58. 6d., in cloth case; postage sd., (in) the lines joining the angles with the middles of minute. The company propose to carry their tube | DUFTY SIMPLE OUTLINES FOR the opposite sides. to the General Post Office. T SCHOOL DRAWING. By JAMES SWETHAN, 1369 (Proposed by * **.)-A telegraph wire The Charimorphoscope, an optical instrument Drawing Master to the Training College. Westminster. This Series of Drawing Copies consists of outlines of capable of sustaining a tension of a pounds, and invented by Mr. H. Treppass, was exhibited by familiar objects, commencing with the most sin weighing 6 pounds to the yard, is stretched, as him in the library of the Royal Institution at a forms. Great care has been taken to avoid ditt tightly as is consistent with its not breaking over recent Friday evening meeting. This apparatus combinations, and the subjects chosen are those art They are of a medium size, a series of posts. to interest children. The posts are generally equally embodies improvements in the construction and graved on steel, and printed on cards. distant from one another, c yards of wire hanging application of the well-known kaleidoscope, the lo Orders, prepaid, may be sent to the Book Depot, between every successive two; but, the telegraph idea of Sir David Brewster, by whom it was per- / Wesleyan Training College, Westminster, S.W. having to cross a river which is more than c yards fected in 1817. In the latter, the beautiful forms Supplied to Schools, at 4s. 6d.; ditto per Post, Ss. wide, a length of d yards of wire hangs between produced are uncertain and temporary ; but in the "Admirably adapted for Schools and Families." Watches the posts on each bank. Show that, in order that Charimorphoscope the effects are entirely inder these two posts inay have no tendency to break, the controul of the operator, who is thus enabled to This day is published, 8vo,cloth boards, prices, each must be inclined to the vertical at all angle produce, in relief, delicate and siunple, or gorgeous DRINCIPLES OF MODERN GEO. equal to and elaborate, patterns, as fancy may suggest. 1 METRY, with numerous Applications to PLANE and SPHERICAL FIGURES; and an APPENDII, E. Mr. Treppass stated that his instrument may thus taining Questions for Excrcise. Intended chirtis be advantageously employed in designing patterns the use of Junior Students. By JOHN MULCAHY, LLE. for silks, carpets, architectural mouldings, icwel- / late Professor of Mathematics, Queen's College, Galway the wire being supposed to be perfectly flexible, llery, ironwork, &c. Second Edition, Revised. and the tops of all the posts to be in the same Dublin : HODGES, SUITI and Co., 101. Grafton Street | Booksellers to the University. London: SIMPEL horizontal straight line. In 12 Numbers, price 1s. each, M.RSIIALL, and Co., 4, Stationers' Hall Court. Soks 1 all Booksellers. 1370 (Proposed by W. J. Miller, B.A., FIRST LESSONS IN DRAWING Mathematical Master, Hudderstield College.)- 1 AND DESIGN; or, Pencilled Copies and Easy WHARTON'S MATHEMATICAL WORKS. Circles are drawn through the angles of a trianóle. Examples. For the Use of Schools and Families, and intended as a preparation for the Drawing Master. By through the three escribed centres, and through GEORGE CARPENTER, Principal Drawing Master in the EIGHTH EDITION, EXLARGED, 12mo, S.; with ANSWERS, 2s.6d., the inscribed and each two of the escribed centres ; Stationers' Company's Grammar School, the Jews' ColT OGICAL ARITHMETIC: being ! show that the radical axes of these circles will lege, &c. &c. The Series embraces a complete course of Elementary U Text-book for Class Teaching, and comment meet the sides of the triangle at the points where Drawing consisting of appropriate and carefully gra- Elementary Arithmetic, with Rules for Mental they are cut by the bisectors of its angles. duated Copies, advancing from the simple stroke to the tical Calculations; and a Course of Fractional and PR most difficult outline, printed in pencil-coloured ink, to I tional Arithmetic, an Introduction to logarithos be first drawn over, and then imitated. The Exercises Selections from the Civil Service, College of Preto have been so simplified, as to render the art of Drawing and Oxford Examination Papers, COLLEGE OF PRECEPTORS. ay easy of attainment as that of Writing. As soon as ANSWERS to the above, separate. Price 6d. II. EIGHTH EDITION 12mo. price is AMPLES IN ALGEBRA FOX out effort, it is confidently believed that no Series of JUNIOR CLASSES. III. SECOND EDITION, 12mo, cloth, price numerous Schools and Families where the books are Mr. J. Reynolds, Dr. Skerry, Dr. White, Mr. employed. DXAMPLES IN ALGEBRA FUN V SENIOR CLASSES; Comprising numerou White, and Dr. E. T. Wilson. CONTENTS:-No. 1. Straight Lines and Combinations The Report of the Examination Committee, in of Straight Lines ;-No. 2. Rectilineal Figures ;-No.3. duated Examples, with the Examination Papers Curvey ;-No. 4. Outlines of Familiar Objects :- No. 5. Parts I.-VI., Is. each, reference to the application of the Board of Mana 1. Shaded Figures:No. 6. Introduction to Perspective: gers of the London Orphan Asylum respecting the THE KEY; containing Complete Sc* No.7. Trees: No.8. Human Figure:- No.9. Animals and I tions to the Questions in the Examples in die periodical examination of their schools by the Rustic Figures ;--No. 10. Ornament;--No.11. Flowers ;No. 12. Maps. for Senior Classes, College, was received and adopted. London : AYLOTT and Son, 8, Paternoster Row. C.F. Hodgsox, 1, Gough Square, Fleet Street, sin th Xo, in Register. COLLEGE OF PRECEPTORS.-AGENCY DEPARTMENT. ADDRESS, 42, QUEEN SQUARE, BLOOMSBURY, W.C. Qualifications. 1603. Classics, Mathematics, French thoroughly, and English subjects. 1043. Classics. Salary 801. to 1001. In Yorkshire. For Easter. A B.A. Lond. Age 22. Salary 801, resident, 1001, non-resident. 1605. English, French, and the elements of Classics and Mathematics. 1062. English, elementary Latin, Greek, and French. Salary 301. to 401. Age 33. As non resident or private Master. 1607. German, French, and Music. A German. Age 33. Non-resident. 1066. Music, Pianoforte, and Harmonium, junior Classics. Salary from 401. 1614. English, Music, French, and Drawing. Age 20. Salary from 301. to 401. A Lady. 1627. Classics and Mathematics. Age 24. A B.A. Cainb. 1628. French, Latin, History, Geography, and Arithmetic. Salary 601. 1068. Classics. Salary to commence at 801. In Kent. 1631. Latin and junior Mathematics, English and French. Age 16. 1072. Junior English Master. Salary 251. In Kent. Salary 101. to 157. 1080. Physical Sciences, Chemistry, and English; Book-keeping desirable. 1638. Drilling or Duty Master. 10 years in the army, and retired as a Salary from 601. to 1001, In the N. district. Serjeant. Age 28. Salary ll. per week. 1081. French, and to assist generally. For Easter. In Yorkshire. 1640. Junior Classics and Mathematies, English and French, Land1082. Non-resident Governess to teach Music, French, Arithmetic, and surveying and Drawing. Age 23. Salary 401. to 501. English. From 25 to 35 years of age. In the W. district. 1615. French and German. Visiting, or Private Pupils. 1083. English, Writing, and Book-keeping. Salary 1001., non-resident. 1677. English, Rudiments of Latin, Greek, and French. Age 18. Salary Hours required in the week, 28. In Yorkshire. 201. to 301. 1084. English, Singing, and Harmonium. Salary 501. For Easter. In 1688. French and German, Classics, Mathematics, Piano, Chemistry, the E. district. English. Salary 601. to 801. Age 25. 1085. Classics and Mathematics. In the S. district. 1697. Classics, Mathematics, French and German, and English. Non. 1086. Junior Classics, Euclid (4 books), Algebra, Mensuration, Land-survey. resident, 1501., in or near Town. A B.A. in Classical Honours Age 29. ing, Mechanics, Book-keeping, and English. Salary 501, Near Bristol. 1700. Classics, French, junior Mathematics, and English. Salary 701. Age 26 1087. English and French (good), junior Latin. 1705. French. Visiting Master or private pupils. 1088. French. In Kent. 1706. English, Writing, Arithmetic, junior French, Mensuration, and 1089. English and French. Salary 501. In Holland. Book-keeping Salary 301. to 401. Age 35. 1090. German. Visiting Master. In the S.W. district. 1708. Elementary Classics, Mathematics, and English. Salary 101. Age 15. 1091. Junior Master, Mathematics, with English. Salary 2016 In Stafford. 1717. English, French, two years in France, rudiments of German, Music, shire. and Drawing. Salary 351. to 401. Age 30. A Lady. 1092. Mathematics. Visiting Master (a Wrangler). In the S.W. district. 1718. Drawing, Writing, and English. Salary 601. Age 23. 1093. A junior Master to teach Drawing and Mapping, elementary subjects. 1724. English, French, Music, Piano and Singing, Rudiments of Latin, In the S. district. German, and Mathematics. Salary 501. Age 27. An A.C.P. A Lady. 1094. Singing. Visiting Master. In the W.C. district. 1728. Mathematics, Latin, and English. Salary 1001., or 1501. uon-resident. Age 26. An M.A. Lond. Univ. A Lady. Age 24. 1731. German, French, and Music. Visiting Master. 1734. English, Arithmetic, Algebra, Euclid (4 Books), Geometrical and ENGAGEMENTS. Perspective Drawing. Salary from 301. to 401. Age 22. 1739. French. Age 33. 1745. French, German, and Drawing. Salary about 301. Age 30. 769, German, French, Spanish, Latin, and Drawing. As Visiting Master. 1746. Classics, Composition (Latin and Greek), French, Algebra, Euclid, 942. French, German, and Elements of Italian and Latin. As Visiting Master. 944. Natural and Experimental Sciences, Mathematics, German, and Drawing. and Arithmetic, English subjects. Visiting or private pupils. 1747. Classics, and English Literature. Salary 801. to 1001.; or private pupils. A German Graduate. Age 33. As Visiting Master. 1748. Classics, Mathematics, and English. 1018. Drawing, Fortification, and Painting. Visiting Master. Visiting, or private pupils. An M.A. 1071. Classics, Mathematics, & general English. M.Ă. Lond. Visiting, or 1749 English, Singing, Arithmetic. Certificated Governess, 2nd class. private pupils. 1210. Drill Master. Salary 251. Age 20. A Lady. | 1750. French, German, and Drawing. Visiting Master. 1212. French German, Classics, and Moral Philosophy. Visiting Master. 1751. Classics, Prese and Verse Composition, Mathematics, Hydrostatics. 1221. Classics, Mathematics, Surveying, Fortification, and French. As Salary 801. to 1001. Age 30. A B.A. Camb. 1752. Classics, Mathematics, French, Hebrew, English subjects. Salary 701. 1239. Piano and Singing. A Lady, formerly a student of R.A.M. Non to 801. Age 32. resident. 1753. English, Music, Singing, Writing, and Arithmetic. Age 19. Salary 1301, Classics, Mathematics, French, and English. Visiting Master, or for about 201. A Lady. private pupils. 1754. English, Writing, Arithmetic, Drawing, Freehand, Landscape, and 1313. Classics, Junior Mathematics, and English. Visiting Master, or for Figure. Age 21. Salary about 301. private pupils. Age 49. M.A. Oxford, and Third Class Classical Honors. 1755. English, Writing, Arithmetic (thoroughly), junior Latin, Euclid (1st 1408. English, French, Music, Singing, Drawing, and Calisthenics. A Lady book). Age 24. Salary 301. to 401. Age 23. Salary 251. 1756. English, Music, French, and the rudiments of Drawing. Age 25. 1447. Drill Master. Salary 301. A Lady. 1484. German, French, Italian, Drawing, and Painting. A German. Age 41. 1757. English (thoroughly), junior Latin, Arithmetic, junior Algebra, and Visiting Master. Book-keeping. Age 44. Salary 601. 1487. Classics, junior Mathematics, and English subjects. Age 35. Salary 1758. Chemical and Mechanical Sciences, Mathematics to Trigonometry, from 601. to 801. Conic Sections, and Classics. Visiting, or private pupils. 1536. French, English, Drawing, and Music. As Visiting Governess. 1759. English, junior Latin, and Algebra, Arithmetic, and the rudiments 1557. English subjects, Use of Globes, Arithmetic, Algebra, Mensuration, of French. Age 24. Salary 307. to 351. Book-keeping, Land-surveying, Trigonometry, Drawing, Navigation, 1760. Classics, Prose and Verse Composition, Mathematics, and Mechanics, Chemistry, Mechanics, and junior Latin. Age 27. Salary from 501.. Age 25. Salary 100!. A B.A. Camb. in Classical Honours. 1564. The Greek, Latin, Sanskrit, or German Language on the principles 1761. French, German, Italian, and Drawing, Classics, and English. Age 28. of Comparative Grammar. Private Lessons, or as Visiting Master. Salary 601. 1762. Mathematics, junior Classics, French. Age 23. Salary 1001. 1763. Junior Classics, Arithmetic, Algebra (junior), English, Writing, and Visiting. French Grammar. Age 39. Salary about 301. 1575. Classics and Mathematics, French and Italian. Visiting, or private 1764. Classics and Mathematics, French and German conversationally. pupils. Graduate of Cambridge. Age 34. Salary 1001. A B.A. Lond. 1580. German and French. A Gerinan clergyman. Age 31. Salary 1765. Classics, French, Arithmetic, Algebra, and Euclid, Book-keeping, and from 501. English subjects. Age..?.. wishes ge 20. Wishes for a private Tutorship in a family. 1590. German, French, and Piano, for beginners. A German. Age 38. | 1766. Mathematics and Classics, with English subjects. Age 25. to 25. a A B.A. Salary 701. Camb., and Prizeman. Private Pupils, or Visiting Master. 1596. High Mathematics, Classics, English, and French. Age 37. Salary 1776. Mathematics, Latin, French, English, and Practical Science. Visiting from 1001.; or for private pupils. Master. No. in |