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MATHEMATICAL QUESTIONS AND

SOLUTIONS.

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... 3x2-= + (2x + 4), 1201 (Proposed by Mr. Matthew Collins.)Without the help of Fluxions, or the differential from which quadratic in x, we obtain Calculus, or any reference whatever to or reasoning about infinitely small quantities; show how to 1158 (Proposed by Mathematicus.)-It is find, by plain common Algebra alone, the value of required to find the point at which a right angled, that will render x-x2, or x-x3, or x3-x3, or triangle must be fixed; so that when its right sin x cos2, a maximum. angle is struck by a blow perpendicular to its plane, it may begin to revolve about an axis SOLUTION BY MISS STEPHENS, OF CAMELFORD parallel to its hypothenuse.

SOLUTION BY MR. STEPHEN WATSON.

In any triangle the line from one of the angles to the middle of the opposite side passes through its centre of gravity; hence if the triangle be fixed at any point in this line, and then struck at the angle by a blow perpendicular to its plane, it will begin to revolve about a line parallel to the side bisected by the line.

1159 (Proposed by Mr. Stephen Watson.)— A semi-ellipse is placed in a vertical position, and a heavy particle is just set in motion at the highest point of the curve, and descends down it by its own weight; required the point at which it will leave the curve, and its velocity at that point.

SOLUTION BY THE PROPOSER.

Let zy be the coordinates of the point where the particle leaves the curve; a, b the semi-axes of the ellipse, b being vertical; v the velocity, and r the radius of curvature, at the point (xy); and the angle which the normal at (xy) makes with the axis of x. Then v2 2g (b-y), and the centri

fugal force at (xy) is

=

PROPRIETARY SCHOOL.
Let x-x=max.
or x2-x--M suppose
x2−x+4=-}-M
x-1=√4-M

x= ± √÷−M

and in order that may be real, M cannot be
greater but may be equal to which is therefore
its greatest value, in which case x=.

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DF (BF)=L+ 2...

Con.-Let L and represent the two given lines, and form the L EBD = given 4, making EB L and BD = 1. Produce BD to F, so that DF L+7; then BD + (1)

=

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but this force is also =g sin Also produce BE to H, so that BH=BF. Bisect Hence 2 gallons must be taken from A and 14
BH in I, and draw IF, and from I apply IM= from B.
IH to meet FE produced; also draw EN MI,

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which determines the point (xy), and the required.. AB+ AC=AB+AE=EB=(con.) L... ... (2). velocity is

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1183 (From Todhunter's Calculus.)-A certain territory is bounded by two meridian circles, and by two parallels of latitude, which differ in longitude and latitude respectively by one degree, and is known to lie within certain limits of latitude; find the probable superficial area.

SOLUTION BY MR. STEPHEN WATSON. Let be the latitude of a place equally distant from the two parallels of latitude; then the two latitudes are 0-1, 0+, and the surface lying between these parallels is,

2r [sin (0+)—sin (0-4)=4πr2 sin cos 0, r being the radius of the earth, also the portion of this surface lying between the meridian circles is, r2 sin cos 9. Hence if a, B be the two latitudes between which the territory is known to lie, the required area is

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It now only remains to show that D is the point
of contact of the escribed circle with the base
BC.

Since FCK is an isos. ▲ ( s), we have
CF=CK (=2CN2AC).......
But (1), BF L+ 2;
.. BC+CF= L +27;
also (3), BC+2AC= L+ 21.
But (2), AB+ AC=L;

.. adding, AB+3AC+BC=2 (L+7),
i.e., AB+3 AC+ (BD+DC) = 2 (L+ 1).
But (2 and con.), (AB+ AC) + BD=L+l;
.. subtracting, 2AC+ DC=L+ 1.
Hence, AB+ AC+BD=2AC+ DC.
And taking AC from each, we get
AB+BD=AC+ DC.

NEW QUESTIONS.

1216 (Proposed by W. J. Miller, B.A., Eltham.)-In the figure to the solution of Ques tion 1175, ("Educational Times" for May) if w be the 4 between the conjugate semi-diameters e, e, and c2. cos xs2 cos p, prove that c. sin 2 =-2ab. cot w, and that

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1218 Proposed by Matthew Collins, A.B., Dublin.)-Prove that a prime number p 2n + 1 will divide 13"-1 whenever p is of the form 13m 1, 13m3, or 13m+4; and when p is not of any of these forms, prove that it will divide 13" +1.

(N.B.- The author of the first correct demonstration of this new theorem, published in the "Educational Times" before the 1st of next September, shall receive the "Educational Times" gratis for one year from the proposer.)

1219 (Proposed by Mr. W. Hopps, Hull.)Divide a right angle geometrically into two parts, such that the tangent of one part shall be equal to

the side of the other.

SOLUTIONS RECEIVED:

Mr. S. Bills.-1191.
Mr. Stephen Watson.-1165, 1169, 1177.

NOTICES TO CORRESPONDENTS :-
We shall give the earliest possible attention to
the Problems sent by Mr. Matthew Collins, B.A.
We trust shortly to have less pressure on our
columns, and we shall then endeavour to carry out
the suggestions of S. W.

ON ABRIDGED NOTATION.
By the Rev. R. H. WRIGHT, M. A., Trin. Coll., Cam-
bridge; Head Master of the Grammar School,
Ashford.

MR. EDITOR,-Having on several occasions contributed my mite towards the Mathematical part of your valuable Journal, I have ventured once more to offer you a short paper on the application of abridged notation to certain geometrical problems. It is well known to most of your readers who may take an interest in this interesting system, "That if any point be taken within a triangle, and lines be drawn through it, from the angular points to the opposite sides, a new triangle will be formed by the intersection of those lines and the sides of the triangle; and if the three sides of the new triangle and the sides of the primary triangle be produced, the three corresponding pairs will meet respectively in three points, which three points will be in a straight line." This property has been discussed by myself in the "Educational Times" years back, and occurs in some of the elementary books on the subject.

DD

For reasons hereafter obvious, I purpose to investigate a particular case of this. For instance, let the new triangle be formed by the feet of the perpendiculars from the angular points on the opposite sides. Let A B C be the primary triangle and D E F the new one. Now, in my former papers I have shown that we have for

EF-a cos A+B cos B + y cos C=O....(1)
DF acos A - B cos B+y cos C=O....(2)
DE a cos A+B cos B-y cos C=O.... (3)
BC a=0

(4) (5)

(6)

AC B=0.... AB yo... Now, if D F and A C be produced to meet, they will intersect in some line whose equation will be represented by

a cos A-B cos B+ y cos C + λ B=0.. (7) The same may be said of D E and A B

a cos AB cos B-y cos C+y=o.... (8) And if we eliminate a from (1) and (4) and substitute in (7) and (s) we get the values of λ and A', viz. :

λ = 2 cos B, λ'=2 cos C

(7) and (8) are identical, and the three intersections are on the same straight line whose equation is

.(9)

a cos A+B cos B+ y cos C=O Again, let there be a second inscribed triangle formed by joining the middle points of the sides of the primary triangle, which we will call D'E'F' Then for

E'F'-a sin A + B sin B+ y sin C=O....(11)
D'F' a sin A-B sin B+ y sin C=O.... (12)
D'E a sin A+B sin B- sin C=O.... (13)
Now for

AP parallel to EF B sin C+y sin B=0) (14)
BQ parallel IF y sin A+ a sin C=O (15)
CR parallel DE a sin B+ B sin A=0 (16)
For the line passing through the intersection of
(12) and (15) we may write

......

....

a sin A-ẞ sin B+ y sin C + λ (y sin A + a sin C)=O . (17) Also, for the line passing through the intersection of (13) and (16)

a sin A+B sin B-y sin C+λ′ (a sin B + B sin A)=0................ (18) Now eliminating 7 and 8 respectively from (11) and (14) and substituting respectively in (17) and (18) we get

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substituting these values of (-A) and (-x') in (17)
and (18) they become identical, that is;

a cos A+B cos B+y cos C=O.... (19)
hence P, Q, R range in the same straight line.
But (19) is identical with (9), hence,-
Those six enunciated points lie in the same
straight line.

The above particular properties appear to be
both interesting and remarkable, for which reason
I selected them as an introduction to the discus-
sion of a more general property proposed, in the
“Mathematician," by my late talented friend Mr.
Weddle, page 209, Vol. II., and solved by another
departed friend and eminent mathematician, Pro-
fessor Hearn, of Sandhurst; and similarly by my-
self, in that same volume. The Proposition runs
thus, "Let lines be drawn from the angles of a
triangle through two points and terminating in the
opposite sides, then if the extremities of each set
of lines be joined, two inscribed co-polar triangles
will be formed. Also from the angular points of
the primitive triangle let lines be drawn parallel
to the corresponding sides of these triangles, and
intersecting those of the other: it is required to
prove that the three points of intersection range in
the same straight line."

Its

aminations. In 1860 the students contributed £544 odd to the funds of the institution. income, however, being almost entirely confined to the payments of the pupils, and to occasional donations, is so inadequate to the necessary expenditure, that the institution must either be closed or re-constituted in a solvent and self-supporting state. The first alternative is greatly deprecated by the noblemen and gentlemen named above, and also by Mr. Gladstone, the Chancellor of the Exchequer, inasmuch as the classes are acknowledged on all hands to have supplied an urgent public want and produced excellent educational and moral results. The object now sought to be attained is to make them self-supporting for the future, and with that view it is proposed to re-constitute them on a more permanent basis and in a collegiate form, the government to be vested in a council, composed partly of a limited number of friends of education, with power to confer associateships and small scholarships on students. The advantages of such an arrangement, it is suggested, would be manifold and various; the tastes, and minds, and feelings of the members would be educated to a higher standard; their opportunities of contact with the best of their fellows for purposes Should an interested reader wish to apply the either of friendship, of study, or of business, would beautiful system of abridged notation to this pro-be much enlarged; and as they profited by their blem, he must bear in mind that if a=0, B=0, Yo, represent the sides of a triangle, then the sides of any inscribed triangle may be represented β 1=0 by a+nẞ+ +3+ly=0, ma+ 1 y=o where l, m, n are constants, and lastly in order that the lines joining corresponding angles of the two triangles may meet in a point we must have the condition l m n = — 1. These useful properties, of which I have made extensive use in a work shortly intended for publication, have been fully discussed by myself in former Numbers of the "Educational Times."

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COLLEGE OF PRECEPTORS.

studies they would acquire titles of distinction and prizes of honour. On the production of a list of young men intending to pay for instruction in the classes, or otherwise to contribute to the funds of +the college, to an extent which may afford a reasonable assurance that an annual income of not less than 7007. may be derived from these sources, it is proposed to raise by donations a fund of 2,000l. to give practical effect to the project. The scheme was earnestly recommended to the meeting last evening by the chairman and by the Rev. Charles Mackenzie, Mr. E. G. Clarke, the Rev. Richard Whittington, Mr. Marshall Carpenter, and other gentlemen. Resolutions were passed to the effect that it was highly desirable that the basis of the present institution be enlarged, its constitution remodelled in the form of a college, with the prito obtain members to the number of 1000 before vileges of associateship, and pledging the meeting the 1st of July. Four of the students at the evening classes took part in the discussion, namely, Mr. E. G. Clarke, Mr. Marshall Carpenter, Mr. Bolton, and Mr. Day. Mr. Carpenter is one of two who have recently obtained public appointments, and Mr. Clarke has gained the Tancred Exhibition at the University of Cambridge for the Study of Law.-The Times.

AT the Meeting of the Council on the 15th of
June; present A. Hill, Esq., V.P., in the chair;
Mr. Andrews, Mr. Bidlake, Mr. Clennell, Mr.
Isbister, Dr. Jacob, Rev. Mr. Jones, Mr. Long,
Dr. Pinches, Dr. Schaible, Dr. White, and Dr.
Wilson; the following gentlemen were elected
Members of the College:-

Mr. S. H. L. A. Bechaux, B.A., Cambridge.
Mr. E. S. Clack, Hackney.
Mr. J. H. Clifft, Bath.
Mr. S. Crew, Great Berkhampstead.
Mr. D. Cruickshank, Southampton.
Mr. J. J. H. Harris, Evesham.
Mr. J. H. Hay, Kennington Oval.
Mr. C. Whitaker, Spalding.

PROPOSED COLLEGE FOR THE CITY OF LONDON.-On the 17th of June, a public meeting, at which Mr. R. N. Fowler, the banker, acted as chairman, was held in Sussex Hall, Leadenhall Street, with the view to promote a scheme recently propounded at a meeting convened at the residence of the Bishop of London, by Earl Granville, Lord Lyttelton, Mr. Henry Hoare, Mr. Harry Chester, Mr. H. Sykes Thornton, and others, for reestablishing the metropolitan evening classes, held for many years at Crosby Hall, upon a new and broader basis. The classes in question, of which with the approval of the late Bishop of London, the Prince Consort is the patron, were established, by the Rev. Charles Mackenzie and others in 1848, and have since been the means of conferring incalculable benefits in the way of education upon

Alphabetical List of the Principal College and
School Books and New Editions published during
the Month ending June 30th.

Bentley (Robert) Manual of Botany, illust., fcap. 8vo.
Binns (Williani) Elementary Treatise on Orthographie
Projection, 2nd edit., 8vo.

Bourne (John) Treatise on the Steam Engine, 5th edit., 4to.
Bullock (Thos. and Fred.) Illustrated History of England
with Questions, fep. 8vo.
Caldwell (Rev. R.) Comparative Grammar of the Dravidian,
8vo.
Campin (Francis) Practice of Hand-Turning in Wood,
Ivory, Shell, &c., sm. post Svo.
Carter (F. H.) Bookkeeping adapted to Commercial and
Judical Accounting, roy. 8vo.
Cheere (Rev. Edward) Church Catechism Explain d. new
edit., fcap. 8vo.

Graduation under the Medical and
Scottish Universities Acts.

Circle of the Sciences (The) Vol. 5, new edit., cr. 8vo.
Clarke (Hyde) New and Comprehensive English Diction-
ary, 2nd edit., corrected, 12mo.
Curtis (J. C.) Elements of the History of England, 18mo.
Cumin (Patrick) Popular Education of the Bristol and
Plymouth Districts, 8vo.
Dower's Popular Atlas for Schools, Families, &c., 12 Maps,

roy. 8vo.

Dunce's Dessert (The); or Horatian Trifles and Ilomeric
Cream, Part 4, to.

Famous Boys, and How they became Great Men, 8th

enlarged edit., fcap. Svo.

Forbes (Duncan) Smaller Hindustani and English Dic
tionary, imp. 16mo.
Jeans (H. W.) Navigation and Nautical Astronomy, Pt.

young men engaged in business in the city. From
1848 to the present time the number of pupils who
have received instruction is about 6000, and since
1856, when the Society of Arts' examinations
began, the pupils of these classes have carried off
138 certificates, many of them of the first class, and
18 prizes; and 10 of them have obtained clerk-Lamborn (Dr. R. H.) Rudimentary Treatise on the Metal-
ships in the public service after competitive ex-

I, new edit., 12mo.

Johnson (Thos.) Græcorum Epigrammatum et Poematum
Delectus, editio nova, 12mo.

lurgy of Silver and Lead, 12mo.

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Miniature French Dictionary, 32mo.

Niebuhr's Tales of Greek Heroes, German Text, Notes by
A. Buchheim, 2nd edit., 12mo.

Roche (Antonin) Du Style et de la Composition Littéraire,
2nd edit., 12mo.

Rogers (E. H.) How to Speak Hindustani, a Guide to
Conversation, 12mo.

Ross (Robert) Manual of English History for Senior Classes
in Schools, cr. 8vo.

Ingov. ITS USAGE AND SENSE IN HOLY SCRIP-
TURE. 2nd Edition. Price 1s. 6d.
Ιησούς, κυριος. THEIR USAGE AND SENSE IN
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AN EXAMINATION INTO THE SIGNIFICATIONS
AND SENSES OF THE GREEK PREPOSITIONS.
Price 6d.

Selectæ Præscriptis, with Translation, by John Pereira, Literal Translations on Definite Rules of Translation, and

13th edit., roy. 32mo.

Strickland (Agnes) Lives of the Bachelor Kings of Eng-
land, cr. Svo.

Struggles (The) of a Village, Lad, new edit., fcap. Svo.
Tegetmeier (W. B.) First Lines of Botany, 18mo.

Weale's Classical Series, Herodotus, T. H.L. Leary, Part
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Williams (Chas. W.) On Heat in its Relations to Water
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Woodward (Charles) Introduction to the Study of Polarized
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Robert Sutton, the Founder of the Charterhouse, and Dean Colet, had as little idea of what London would be one day, as we have of what it will be three hundred years hence. Every facility which money could give might safely be reckoned upon, for the present sites of public schools in London are of enormous money-value. They are literally built upon gold. Not so very long since the sum of 200,000l. was offered for the Charter House site. It was intended to convert it into a central railway terminus. Unfortunately the offer was refused. Look again at St. Paul's School. When one remembers the prices which were offered and demanded for that little speck of ground which com- A LETTER TO THE RIGHT REV. DR. WISEMAN manded the south-eastern view of the Cathedral, it seems almost profane to offer even a guess at the value of well-nigh one side of St. Paul's Churchyard. The "Mercers," who are the guardians of Dean Colet's will, might, upon the annual interest of the difference in value between a London and a country site, almost undertake to convert the day boys into boarders, and to find them in beef, lodging, and clothing, as well as Latin and Greek.

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in his dress. For the poor the cloth was merely DE PORQUET'S FRENCH EDUCA- of Straight Lines - No. 2. Rectilineal Figures :-No. 3.

chequered with another colour, as in the shepherd's plaid. For the superior, breachans, green and black, with a red stripe, seem to have predominated. The breachan, or plaid, was originally a large mantle of one piece, belted round the body. Some are of opinion it was also called the tringhas, or truis, from the word "truss," to tuck up, and that it formed the whole dress (as the drapery of the ancients consisted but of one piece of cloth), and that the word truis was transferred to the pantaloons and stockings joined, when the ancient dress was prohibited. The word "kilt" is lowland Scotch. It has much the same meaning as truis; for "to kilt" is to tuck up. In Gaelic, the kilt is feile-beag; and the time of its separation from the ancient feile-beag into a waistcoat and kilt is uncertain, as is also the adoption of the sporan, or

pouch, so remarkable a feature in the Highland costume.-The Ladies' Treasury.

Mr. J. N. Smith, S.C.L., Oxon, Principal of Marlborough House School, Brompton, was presented, at a meeting of the former pupils of that school, on the 20th April, with a handsome silver tea service, as an expression of their esteem and gratitude.

Mr. W. T. Miller, B.A., formerly of Eltham, Kent, has been appointed Mathematical Master in Huddersfield College.

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CONTENTS:-No. 1. Straight Lines and Combinations Curves;-No. 4. Outlines of Familiar Objects;-No. 5. Shaded Figures;-No. 6. Introduction to Perspective;No. 7. Trees;-No 8. Human Figure;-No. 9. Animals and Rustic Figures;-No. 10. Ornament;-No. 11 Flowers;— No. 12. Maps.

London: Aylott and Son, 8, Paternoster Row.

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SEVENTH EDITION, ENLARGED, 12mo, 2s.,

LOGICAL ARITHMETIC; being a

Text-book for Class Teaching, and comprising Elementary Arithmetic, with Rules for Mental and Prac tical Calculations; and a Course of Fractional and Propor tional Arithmetic, an Introduction to Logarithms, or Selections from the Civil Service, College of Preceptors, and Oxford Examination Papers. By J. WHARTON, M.A., M.C.P.

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COLLEGE OF PRECEPTORS.-AGENCY DEPARTMENT.
ADDRESS, 42, QUEEN SQUARE, BLOOMSBURY, W.C.

REGULATIONS.

1. Applicants for vacant situations must send to the Secretary a statement of their qualifications, two copies of their testimonials, the names and addresses of their employers, if any, during the preceding three years, the salaries required, and their own addresses. The fee for insertion in the Register is one shilling.

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3. The requirements of Principals of Schools, Members of the College, are inserted in another Register on payment of one shilling: the fee to non- Members is kalf-a-crown. These fees cover all charges for correspondence, which will, if required, be undertaken by the Agency Department, and for Advertisements. 4. Every facility is afforded for interviews between Principals and Assistants at the College Offices. **All communications must be addressed to the Secretary, 42, Queen Square, W.C.

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476. Junior Assistant to teach English Subjects, with Elementary Latin and French. For Midsummer. At Brighton.

488. Writing Master. For Midsummer. Near Town.

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506. English Master, age about 22. Salary 307. In Worcestershire. 508. Junior Mathematics, with English Subjects and Drawing. Salary] 201., to 301. In Herts.

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Salary 807., Salary 357.

517. French, and general Subjects. Salary 607. In Kent.
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520. French, Latin, Junior Mathematics, and English Subjects.
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521. (i.) Classics and Mathematics, Mechanics, Mensuration and LandSurveying. Salary 501.

(ii) Junior Assistant; good Penman. Salary 257. to 30%. Near Bristol. 522. English Master. Salary from 35l. to 40l. In Guernsey. 523. General English Master. Salary about 401. Near Town.

524 Physical Science, especially Chemistry, and the usual branches of English, with Book-keeping. Salary 601. to 1001. Near Town.

525. English Master. Non-resident. Salary 701. to 901. In Liverpool.

526. Junior Assistant. Salary 401. In Brighton.

527. (i.) French, German, and Music. Salary 551.

(ii.) English, Writing and Arithmetic. Salary 50%. In Yorkshire. 529. Good general English Master. Salary 351. to 401. In Hants. 530. English Master. Salary from 351. to 451. In Town.

531. (i.) English, Mapping, Land-Surveying, Writing, and Book-keeping. Salary 357.

(ii.) French and Drawing. Salary 351. In Kent. 532. English Master. Salary 50l. In Belfast.

533. (i.) English Master. Salary 251.

(ii.) Latin and French. Salary 50%. In Hants.

535. (i.) English, Junior Latin, and Drawing. Salary 201.

(ii.) Articled Papil, who would be prepared for the Matriculation examination of the University of London or for the Cambridge or Oxford Middle Class examination. Premium for 3 years. 501. In Worcestershire. 536. Latin and English Subjects. Salary 30l. to 35l. In Kent.

737. English Master for the Commercial department in a Grammar School. Salary 50. In Lancashire.

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97. Model Drawing, Figure, and Landscape, &c. (Student of the Royal Academy.) Age 26. Non-resident.

224. Classics and Mathematics, especially the latter. Accustomed to Teaching. B.A. of London. Non-resident.

340. Painting and Drawing. A Member of the Institute of the Fine Arts. One guinea for each attendance within a reasonable distance of London. 430. Drawing and Painting. Age 22. A Certificated and Prize Medallist of the Department of Science and Art. As Visiting Tutor.

443. Mathematics, Junior Classics, French, Drawing, and Writing. Age 34. As Morning Visiting Master.

451. Visiting Teacher of French, Mathematics, and Writing.

497 High Classics. Age 30. A graduate of Oxford in Holy Orders. 559. English, Music, Drawing, and Painting. As Morning Governess in a School or Family. Age 21. Salary 401., non-resident.

562. German, French acquired in Paris, Pianoforte and Drawing for Juniors. Age 26. As non-resident or Visiting Master.

639. Classics and Mathematics, French and Italian. Age 52. As resident or non-resident Master.

647. German and French. As Visiting Master.

649. Classics, Mathematics, and English subjects. Private pupils. 650. French Language and Literature. As Visiting Master from 1 till 9. 672. Mathematics, Classics, French, and Geometrical Drawing. Graduate in Mathematical Honours of T. C. D. Age 33.

683. Classics, Mathematics, French, and English. Age 22. Salary 70l. to 801. 684. High Mathematics, pure and mixed, French, German, Chemistry, and Drilling. Age 28. Educated at St. Cyr. Visiting or non-resident Master. 688. English, Junior Latin and Mathematics, with Elementary French. Age 20. Salary 251.

690. English, French, Classics, and Mathematics. Age 25. Salary from 751. to 80l. non-resident. 691. Highest Mathematics. To teach twelve or eighteen hours per week.

A Wrangler, 1861.

692. Highest Mathematics, with Classics, &c. A Wrangler, 1861. Requires an engagement at Midsummer for about three months.

696. Classics and Mathematics. Graduate in Classical Honours of Aberdeen. Age 25. Salary 701.

697. English, Music, French, German, and Drawing. A Lady. Age 30. Salary 601.

698. English Subjects, French, German, and Music. A Lady. Age 19. Salary 157. to 201.

699. Elementary Mathematics, English, French, and Latin. Salary 201. 700. German and French thoroughly, with Classics and Mathematics. A German. Age 32. Salary 501.

701. French and Classics. B.A. of Univ. of Paris. Age 33. Salary 501. 708. Highest Classics, Junior Mathematics, French and German, Age 22. Salary from 1507. to 2007. Non-resident.

709. Junior Latin and Mathematics. Age 22. Salary 301. 710. German, French, and Piano. Age 30. Salary 607. to 70l. 712. Classics, French grammatically, Pencil and Water-colour Drawing, and moderate Mathematics. Age 25. Salary 807.

713. English Subjects, Music, and the Rudiments of Drawing. A Lady; age 23. Salary according to duties required. 714. Classics and Mathematics. Age 37. Salary 607.

No. in Register.

THE EDUCATIONAL TIMES.

College of Preceptors.-Agency Department continued.

Qualifications.

715. English Subjects, Mathematics, Navigation, Land Surveying, Latin, Elementary French, Violin, and Piano. Age 23. Salary 607.

716. Classics and Mathematics, Junior French. Age 23. Salary 70l. to 801.
717. English, Arithmetic, Writing, and Junior Latin. Age 21. Salary 257.
718. German, French, and Italian. Age 50. Salary 601.

719. Latin, History, English, Junior Mathematics. Age 21. Salary 401.
720. English, History, Junior Mathematics. Age 21. Salary 401.
722. German thoroughly. Age 30. Salary 1007.

724. French. B.A. of the University of Paris. Age 31. As Visiting Master.
725. Mathematics, pure and mixed, French, German, Practical Chemistry,
Fortifications, and the Subjects of Military and Civil-Service Exami-
nations. Salary 1007. resident, 1507. non-resident.
726. Drill Serjeant. Age 30. 4s. per lesson.

727. French, English, and Junior Mathematics. A native of Paris. Age 25.
Salary 701.

A Doctor of Laws of the Univ. of Paris. As Visiting Master, or Private Tutor.

728. French and Italian.

729. Junior Assistant, English Subjects, with Junior French. Salary 251.

730. French, Mathematics, and Natural Philosophy. A Pole. Salary 501.

731. French. A Bachelier-ès-Lettres. Age 30. Salary 401.

Age 26.

Age 32.

732. Classics and Junior Mathematics, Modern History, and Geography. B.A. of Oxford. Age 25. Salary 1007.

733. Highest Mathematics with Classics. B.A. of Lond. Age 28. Salary 150l. to 2007. non-resident.

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734. Classics, Mathematics, and French. Age 45. Salary 60l. to 70%.
735. Classics and Mathematics, especially the latter. B.A. and Mathematical
Scholar of Univ. of London. Requires an engagement during the long
vacation.

737. Classics, Greek and Latin Versification, Mathematics, Elementary
Statics and Dynamics, French. B.A. London; first in Classical Honours.
Age 34. Salary from 1207.

748. Music (thoroughly), Singing, English generally, in a School. A Lady. Age 22. Salary 301.

739. English, French, and German. Age 39. Salary 30l. to 4017.

740. French, Drawing, Fortification, Mathematics, Surveying, Painting in Water Colours. Salary 100l. resident, non-resident preferred.

741. Mathematics, Elementary Classics, and Drawing, Vocal Music. Age 35. As Visiting or Non-resident Master.

742. French, Rudiments of German, and English thoroughly. Salary 50%.

Age 31.

742 A. English, Middle Classics, Junior Mathematics, Rudiments of French Music. Age 20. Salary 50l. to 60%.

743. French, German, Mathematics, Drawing, and Music. Salary 1207. 744. Writing Plain and Ornamental, Drawing, Water and Oil Colour Painting. Age 26. Salary 557. to 60l., non-resident.

Age

745. Classics, French, German, and Italian, and Junior Mathematics.
33. B.A. Oxford. Salary, resident, 1007.; non-resident, 1807.
746. Latin, Elementary Greek, Algebra, French, and English. Age 21.
Salary from 25l. to 30l.

747. Classics and Mathematics. B.A. Cambridge. Age 24. Salary from
907. to 100%.

748. French. Age 30. Salary 501.

749. Latin, Greek, English, Euclid, Algebra, Elementary French. Age 19. Salary 351.

750. French, Gymnastics and Drilling. Age 36. Salary 401.

751. French, German, Mathematics, Fortification, History, Geography. Age 37. As Visiting Master.

752. Engish, Junior Latin, and Junior Mathematics. Age 19. to 351.

Salary 301.

753. Middle Classics and Mathematics, Book-keeping, Mensuration, English and French grammatically. B.A. Queen's University, Ireland. Age 25. Salary 501.

754. Junior Assistant. Age 20. Salary from 251.

756. French, German, and Italian, Classics and Mathematics, Natural Sciences, Music, Drawing, Drilling and Gymnastics. Age 32. Salary from 801. resident; non-resident preferred.

757. Landscape and Figure Drawing, Painting in Oil and Water Colours, Fortification and descriptive Geometry. As Visiting Master.

758. Lectures on Natural Philosophy and Chemistry. Age 28. As Visiting Master.

759. French and Drawing, English and Arithmetic. Age 23. Salary 401. 760. English, Latin, French, and Junior Mathematics. Age 22. Salary

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[July, '61.

762. English Subjects, and the elements of Latin. Age 17. Salary 107.
with time for self-improvement.

763. English Subjects, Junior French, and Latin. Age 207. Salary 30l. to 401.
765. English Subjects, with Junior Latin. Salary 601.
766. Junior English Master. Age 26. Salary 20l. to 251.

767. Mensuration, Land Surveying, Junior Mathematics, Writing Plain
and Ornamental, Drawing, Mapping, and Junior Latin. Age 21. Salary 351.
768. Classics, Mathematics, French, Italian, and English Subjects. As
Visiting Master.

769. German, French, Spanish, Latin, and Drawing. As Visiting Master.
770. Mathematics, Mensuration, and Junior Classics.
Salary 50%.
M.A. Cambridge.

771. French, Latin and English. A Frenchman. Age 38. Salary 70l. to 80l.
772. Mathematics, French, Music, Junior Latin, and English. Age 26.
Salary 601.
773. Classics and Middle Mathematics, English Literature, French gram-
matically, Chemistry theoretically. Salary 1207.
774. Music, French and English Subjects. A Lady. Age 18. Salary 251.
775. Latin, German, French, English, Mathematics, and Drawing. Age 28.
Salary 601.

778. Classics, French, Mathematics, Mensuration, and English. Age 42.
776. English, and Junior Mathematics. Age 21. Salary 30l. to 401.
777. Latin, French, English, and Mathematics. Age 20. Salary 457. to 50%.
Salary 601. to 70%.

779. Junior Classics, and Mathematics. Age 16. No salary required, but
time for self-improvement.

780. French, Elementary German, and Landscape Drawing. B.A. of the University of Paris. Age 37. Salary 50l. to 60l. resident; 80l. to 1007., non-resident.

781. French, Elementary Music, Latin, and Greek. B.A. of Paris. Age 22. Salary 30%.

783. Classics. M.A. of Aberdeen. Age 28. Salary 707.

782. Classics, English, and Junior Mathematics. M.A. of Aberdeen. Age 22. Salary 80%.

784. English, Junior Latin, French, and Mathematics. Age 30. Salary 401.
785. Classics, Prose and Verse Compositions, Mathematics, Mechanics,
Hydrostatics, English, French, and Italian. As Visiting Master.

786. Junior Classics and Mathematics, with the Natural Sciences.
787. French, and Middle Classics. B.A. of Paris. Age 21. Salary 50l.
788. Elementary Mathematics, and Classics. Requires an engagement for
the Long Vacation.

789. Middle Classics, and Mathematics, French and Drawing. B.A. of
Glasgow. Age 23. Salary 701.

790. Latin, Middle Mathematics, Mensuration, Land-Surveying, and English
Subjects. Age 22. Salary 407.

791. French, German, Music, and Drawing. Age 27. Salary 60%.
792. Classics, Mathematics, and English Subjects. Sen. Op. 1851. Age 26.
Salary 1507., non-resident.

795. French, English, Latin, Junior Mathematics, Elementary Drawing,
Mapping, Writing. A Frenchman. Age 26. Salary from 60l. to 70l.
796. English and German thoroughly; the rudiments of French and Music.
A Lady. Age 21. Salary 251. to 301.

797. Classics, Mathematics, Book-keeping, French and English Subjects. Age 40. Salary from 80l. to 1007.

798. German, French, Drawing, and Music. Age 28. Salary from 401.

to 50l.

800. Classics, Mathematics, and English Subjects. Non-resident preferred. Salary 1007.

801. High Classics, Moderate Mathematics, and English. Age 24. Salary from 70l. to 80%.

802. French, Spanish, Drawing, and the elements of German. Age 30. Salary 401. In Town.

803. English, Junior French and Latin, with Drawing. Age 22. Salary 301. 804. High Classics and Mathematics. Salary 1207., non-resident; resident, 907.

806. Lecturer on Physics, Chemistry, Botany, Geology, with Moderate Mathematics.

805. General English Subjects. Salary 201.

807. English and Music. A Lady. Age 18. Salary nominal.

808. Middle Classics, Mathematics as far as Conic Sections, French and
English. Age 24. Salary 50l. to 607.

812. French and German, A Lady, as Visiting Governess.
811. Junior Classics, English, French, Drawing, Book-keeping, and Land-
Surveying. Age 27. Salary 701.
813. French, Drawing, Junior Latin, and English Subjects.
Salary 501.

Age 35.

Printed and Published, monthly, by CHARLES FRANCIS HODGSON, Gough Square, in the Parish of St. Bride, in the City of London; and sold by W. Aylott and Son, 8 Paternoster Row; and W. Wesley, 2 Queen's Head Passage, Paternoster Row. JULY 1, 1861.

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