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The secant increases with the increase of the angle from 1 its value when the measure of the angle is 0 to ∞ when the angle increases to 90°; while the angle increases from 90° to 180°, its secant is negative and decreases numerically from ∞ to -1; then as the angle increases from 180° to 270°, its secant is still negative and increases numerically from 1 to co; and while the angle increases from 270° to 360°, its secant is positive and decreases from ∞ to 1.

The changes in the sign and magnitude of the cosecant, as the angle increases from 0 to 360°, may be similarly traced, or the variations of the secant and cosecant may be traced through those of their reciprocals, the cosine and sine.

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SUMMARY SHOWING THE VALUES OF CERTAIN RATIOS.-(Con.)

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1

The Royal Geographical Society.

THE disinclination of public schools to send in competitors for the Society's annual examination has been made manifest from the decline in the number of pupils examined from year to year, rendering the competition of little importance, and causing the discontinuance of the examination last year. The Society are as interested as ever in the spread of geographical knowledge, and we are informed by the Times that "the Society have decided to appoint for one year an inspector to inquire thoroughly into and report upon the state of geographical education at home and on the Continent. In addition to studying the best method of geographical teaching-chiefly probably in Germany and Switzerland-he will be required to collect and

report upon the best text-books, maps, models, and appliances. The Society will, we understand, offer £250 for the services of a competent inspector, to include travelling expenses, but not the purchase of books, &c., which will be supplied by the Society on the selection being approved by the council. The main business of the inspector will really be to inquire into the methods of teaching on the Continent and reporting thereupon. The actual inspection could probably be accomplished in a couple of months. We need not point out the great importance of the step taken by the Society, and we trust they will be fortunate enough to meet with a competent man."

SOCIETY OF APOTHECARIES.

EXAMINATION IN ARTS.

GEOMETRY.

(Answered by an LL.B.)

1. (a) When is one straight line said to be perpendicular to another straight line?

(b) Draw a straight line perpendicular to a given straight line of unlimited length from a given point without it.

(a) "When a straight line, standing on another straight line, makes the adjacent angles equal to one another, each of the angles is called a right angle, and the straight line which stands on the other is called a perpendicular to it." (I. Def. 10.) (6) Euclid I. 12.

2. Prove that if from the ends of a side of a triangle there be drawn two straight lines to a point within the triangle, these together shall be less than the other two sides, but shall contain a greater angle. Euclid I. 21.

3. Define an isosceles triangle and a right-angled triangle.

Draw a right-angled isosceles triangle.

On a given straight line describe a right-angled isosceles triangle.

"An isosceles triangle is that which has two sides equal." Def. 25.

A right-angled triangle is that which has a right angle. Def. 27.

Take a right line of any length, and at one of

its extremities draw another right line perpendicular and equal to it. Join the extremities of these lines. A right-angled isosceles triangle is formed, and it is also described on a given right line. The last part of the question is therefore answered; but it is probable the examiner intended that the given right line should be the hypotenuse of the triangle in that case. Bisect the given right line (I. 10), and from the

point of bisection erect a perpendicular (I. 11) equal to half the line; join the extremity of this line with the extremities. of the given line. The triangle required

is formed.

4. Define parallel straight lines. Draw a straight line through a given point parallel to a given straight line.

"Parallel straight lines are such as are in the

same plane, and which, being produced ever so far both ways, do not meet." Def, 35. Proposition I. 31.

5. A polygon has 52 sides. To how many right angles are all its interior angles together equal? By Corollary 1 to Proposition I. 32, "all the interior angles of any rectilineal figure, together with four right angles, are equal to twice as many right angles as the figure has sides." Deducting four right angles, therefore, from twice 52, we obtain 100 right angles.

6. What is a parallelogram?

Prove that parallelograms upon the same base and between the same parallels are equal to one another. "A parallelogram is a four-sided figure, of which the opposite sides are parallel." Proposition I. 35.

7. Prove that of all parallelograms which can be described upon the same base and between the same parallels, the one which has the least perimeter is rectangular.

If a rectangular parallelogram be drawn, and

also one with an obtuse or acute angle, the two parallelograms have one side common, and also the sides opposite to the common side are equal to each other, being each equal to the common side; but of the sides which lie between the parallels those which are at right angles to them are the least. (See Euclid, Book I., Proposition 19.) Consequently the perimeter of the rectangle must be less than that of any other parallelogram.

Note-The perimeter is the sum of the lines which bound a figure.

8. Prove that if a straight line be divided into any two parts the rectangles contained by the whole line and each of the parts are together equal to the square on the whole line.

Euclid II. 2.

9. If the sides of a triangle be as the numbers 3, 4, and 6, show whether it will be acute angled or obtuse angled.

The greatest angle must be opposite the side represented by 6 (I. 18). If this angle be acute, the square of the side opposite. it must be less than the sum of the squares of the other two sides (II. 13). If it be a right angle the square of the same side must be equal to the sum of the squares of the other two sides (I. 47); but, since 36 is greater than 25, the angle must be obtuse.

10. Describe a square equal to a given rectilineal figure. Euclid II. 14.

PUBLIC SPIRIT.-An Essay.

Or all places in the world, London ranks first in the quality known as public spirit. Many of the London merchants, of the past as well as of the present day, in a most unselfish manner, left large fortunes, accumulated by the unflagging application to business of half a lifetime, to useful public institutions, either for charity, the advancement of education, or for some similar praiseworthy object. In more recent times it has become customary, when money is wanted for some public object, either to open a subscription list or to receive from some prince merchant an amount adequate to meet the pressing need. The former may be termed national public spirit, the latter individual public spirit. One of the commonest examples of national public spirit arises when famine or any dire calamity attacks many of our fellow-countrymen, or indeed any of our fellow-creatures in any part of the globe. In such cases the Lord Mayor is accustomed to open a fund to which all who can well spare cash subscribe. A committee is mostly appointed to superintend the affair, and to disburse the money in accordance with the intentions of the parties who have subscribed. Again, after any of our "necessary little wars," a voluntary fund is raised for the benefit of the widows and orphans of our sailors and soldiers who may have fallen in battle. The enormous amount which is frequently raised in this way affords the very best example of national public spirit; for, as a rule, there can be no selfish motive influencing the parties, many of whom send their subscriptions anonymously to avoid being pestered by needy folk, who always scan such lists from a selfish point of view. Individual public spirit has of late years had many noble examples. It takes many forms, and benefits the public in a more lasting way, by establishing here a blind asylum, there building and endowing a church in a crowded, godless district, and in a third locality renovating a cathedral or erecting a charity school. Public spirit is by no means confined to this large metropolis. We have had the princely fortune of that big-hearted and noble-minded man Peabody to erect dwellings both in London and New York for the working classes. This man's benevolence deserves the more particular attention that he did not confine his good deeds to his own countrymen, but allowed ours the privilege of an equal participation. Many of our own manufacturers have in the provinces bestowed public parks on towns for the recreation of the inhabitants. Among those who also deserve special mention outside Great Britain are the late Sir Benjamin Lee Guinness, a Dublin brewer, who renovated St. Patrick's Cathedral in that city, at an expense of £150,000; and Mr. Roe, a Dublin distiller, who rebuilt the

ancient structure, Christ's Church, expending little less than the figure written above. Were we to enumerate all the persons who have acted in a highly praiseworthy manner and with public spirit in this country, our essay would extend much beyond the usual limits; but before concluding, it may be as well to mention that not a few of these persons have received a public recognition of their services from the Crown in the shape of a title, and hence there may not be, after all, so much benevolence and disinterestedness in such gifts as appear on the surface.

Telegraphists and Telephonists.

A contemporary says:-It is now thirty years ago that women were admitted into the telegraph office. The story goes that the Queen herself has done much to promote their admission. General Wylde, one of the directors of the Electric International Telegraph Company, and in 1853 equerry to Prince Albert, is said to have suggested the idea to her Majesty, who at once entered into the plan with much sympathy. The experiment was begun with only a few girls, but their number, since in 1870 the telegraphs were bought up by the Government, and even more since Mr. Fawcett has become PostmasterGeneral, so rapidly increased, that at present no less than 659 women are employed at the Central Telegraph Office in London, and about 350 more at the different telegraph and postal branch offices of the metropolis. At Liverpool, Glasgow, Manchester, and other provincial centres, a considerable number earn their living in this branch of the Civil Service, and besides these there are 900 women similarly employed outside the Civil Service.

It is far from difficult for a girl to qualify herself for the post of telegraph clerk. Preliminary competitive examinations, open to one and all, are twice a year held at the General Post Office, London, for which applications are made to the Civil Service Commissioners, Cannon Row, Westminster. Any girl between fourteen and eighteen years of age may be entered for the examination, the fee for which is one shilling. If she satisfies the examiner in dictation, handwriting, and easy sums in the first four rules, she enters the Post Office Telegraph School, where no charge is made for the instruction. Threo months generally qualify the learner for the lowest stage of the actual work in a telegraph office, when she receives a weekly payment of ten shillings. The first month, however, is considered probationary work, during which, if it is seen that a girl has not the necessary qualification, the appointment is cancelled. This period of probation having been passed successfully, the pay begins to rise by degrees to 27s. per week.

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a rule, women do not rise higher, which is attributed to the fact that women do not enter into the work with a thoroughly business-like spirit. "When a young man begins work," said Mr. Fischer, the Controller of the Central Telegraph Office, "he knows that his career lies before him, and that according to his exertions he will rise.

Girls do not seem to see the work

in that light; they only regard it as a means of earning their living, until after a few years they can get married." Some, however, form a laudable exception, and are promoted to higher classes, according to merit. Of the women at the Central Telegraph Office, 424 are second-class telegraphists, receiving weekly pay of 10s., 12s., and 14s., rising by 1s. a week per annum to 17s., and then by 1s. 6d. a week per annum to 27s. Of first-class telegraphists there are 192, with a salary of 288., rising by 1s. 6d. a week per annum to 34s.; 15 assistant supervisors receive £90, and rise by £5 per annum to £110; 15 others receive a salary of £110, rising by £6 per annum to £140; the 8 supervisors receive £150; 6 of these rise by £8 per annum to £180, the 2 remaining ones to £225. The matron receives £150, rising by £10 per annum to £250.

The daily work of the female telegraphist extends over eight hours, within the period of from eight in the morning to eight at night. If they work ten hours per day one week they only work six hours per day the week after. No telegraphist is allowed to leave the premises between the working hours, but as dinner is served in the spacious diningrooms, two of which belong to the women and two to the men, and as breakfast and tea are provided for the whole staff employed at the General Telegraph Office, this is no troublesome regulation. The tea is brought round to the desks by a special staff of twentyfour female servants; the cost of this afternoon cup of tea, with a couple of slices of bread-andbutter, amounts to no less than £1,500 per annum, Breakfast for the early comers and tea are provided free of cost. It occurs occasionally that the female telegraphist has to do Sunday work; night-work she is never expected to do, "for," reasons the head of the institution, "no respectable mother would like her daughter to be out alone after eight o'clock at night or before eight in the morning, and consequently we should have a lower class of girls than is at present the case."

To judge from appearances, the present female staff are girls and women of a very superior class. They work in large rooms, at present mixed with the men; sometimes a single female telegraphist amid a number of men, sometimes vice versa. All are busily intent on their work. Neatly attired, often with a vase of spring

flowers by their side, there is an air of vigour and sprightliness about them which harmonises well with the busy scene of which the women telegraphists form by no means the least attractive feature. But notwithstanding bright looks and rosy cheeks, the report of their state of health as compared with that of their male colleagues is not very satisfactory.

Three years ago the United Telephone Company began to employ women. Preference is given to the daughters of professional men; this, however, is no strict rule. There are no preliminary examinations to be passed before admittance into the telephone office. The different ranks are the clerk in charge, who has the supervision of a room, in which are employed from six to fifteen clerks; the deputy of the clerk in charge, the first and second-class operator, and the learner. According to these ranks the payment varies; as a rule, however, 11s. to 16s. is the weekly pay, which sometimes rises to £1, and in rare cases to 25s. The hours of work are from nine till six or ten to seven respectively.

BEAUTY AND BRAINS.

It is a circumstance which must frequently have struck the observation of the general student that intellectual power and physical symmetry are often closely allied in the same person; indeed, that this is a rule contradicted by but few authentic exceptions. As a general rule, the fashion of the outer casket of a great intelligence is pretty generally of a design worthy of the treasure it contains, and men upon whom destiny has bestowed genius have, as a rule, been additionally favoured by the gift of outward comeliness. Beauty unaccompanied by any superabundance of brain is common enough; the treasure house of art is rich in forms and faces whose loveliness is their one hold upon remembrance; but among the loveliest there are those beneath which some deathless name in art or song has place. A brief and casual glance through our mental picture gallery is sufficient to prove this thesis. Michael Angelo's face, before the symmetry of his nose was spoilt by the fist of his jealous brother artist, must have been a type of beauty as superb and uncommon as was his genius. Dante possessed a countenance truly remarkable for its rugged majesty; and Shakespeare, the intellectual monarch of the world, had few equals indeed in personal comeliAndrea del Sarto, "the faultless painter," possessed a face as flawless as his art; and where among the countless dreams of manly beauty which painters have embodied for us shall wo

ness.

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