of about equal merit bracketed, and that day was devoted to arranging the men whose names appeared in the same bracket in their proper order. A man rarely rose above or sunk below his bracket, but during the first hour he had the right, if dissatisfied with his position, to challenge any one above him to a fresh examination in order to see which was the better. At nine a second list came out, and a candidate's power of challenging was then confined to the bracket immediately above his own. Fresh lists revised and corrected came out at 11 a.m., 3 p.m., and 5 p.m. The final list was then prepared. The name of the senior wrangler was announced at midnight, and the rest of the list the next morning. The publication of the list was attended with great excitement.

About this time, circ. 1772, it began to be the custom to dictate some or all of the questions and to require answers to be written. Only one question was dictated at a time, and a fresh one was not given out until some student had solved that previously read—a custom which by causing perpetual interruptions to take down new questions must have proved very harassing. We are perhaps apt to think that an examination conducted by written papers is so natural that the custom is of long continuance. But I can find no record of any (in Europe) earlier than those introduced by Bentley at Trinity in 1702 (see p. 81): though in them it will be observed that every candidate had a different set of questions to answer, so that a strict comparison must have been very difficult. The questions for the Smith's prizes continued until 1830 to be dictated in the manner described above. Even at the present time it is usual to dictate the mathematical papers for the baccalaureate degree in the university of France, but all the questions are read out at once.

In 1779 the senate-house examination was extended to four days, the third day being given up entirely to moral philosophy; at the same time the number of examiners was increased, and the system of brackets recognized as a formal part of the procedure. The right of any M. A. to take part in it, though

continuing to exist, was much more sparingly exercised, and I believe was not insisted on after 1785. A candidate who was dissatisfied with the class in which he had been placed as the result of his disputations was henceforth allowed to challenge it before the examination began. This power seems to have been used but rarely; it was however a recognition of the fact that a place in the tripos list was to be determined by the senate-house examination alone, and the examiners soon acquired the habit of settling the preliminary classes without much reference to the previous disputations.

In cases of equality the acts were still taken into account in settling the tripos order; and in 1786 when the second, third, and fourth wranglers came out equal in the examination a memorandum was published that the second place was given to that candidate who in dialectis magis est versatus, and the third place to that one who in scholis sophistarum melius disputavit.

In 1786 a question set to the expectant wranglers which required the extraction of the square root of a number to three places of decimals is said1 to have been considered unreasonably hard.

The only papers of this date which as far as I know are now extant are one of the problem papers set in 1785 and one of those set in 1786. These were composed by William Hodson, of Trinity (seventh wrangler in 1764, and vice-master of the college from 1789 to 1793), who was then proctor. The autograph copies from which he gave out the questions were luckily preserved, and have recently been placed in the library of Trinity. They must be almost the last problem papers which were dictated, instead of being printed and given as a whole to the candidates.

1 See Gunning's Reminiscences, vol. 1. chap. III. Note however that the Reminiscences were not written till 60 or 70 years later; and this statement only represents the author's recollections of the rumours of the time. There are reasons for thinking that the statement is exaggerated. 2 The Challis Manuscripts, III. 61.

The paper for 1785 is headed by a memorandum to warn candidates to write distinctly and to observe that "at least as much will depend upon the clearness and precision of the answers as upon the quantity of them." The questions are as follows.

1. To prove how many regular Solids there are, what are those Solids called, and why there are no more.

2. To prove the Asymptotes of an Hyperbola always external to the Curve.

3. Suppose a body thrown from an Eminence upon the Earth, what must be the Velocity of Projection, to make it become a secondary planet to the Earth?

4. To prove in all the conic sections generally that the force tending to the focus varies inversely as the square of the Distance.

5. Supposing the periodical times in different Ellipses round the same center of force, to vary in the sesquiplicate ratio of the mean distances, to prove the forces in those mean distances to be inversely as the square of the distance.

6. What is the relation between the 3rd and 7th Sections of Newton, and how are the principles of the 3rd applied to the 7th?

one.

7. To reduce the biquadratic equation x+qx2+rx+s=0 to a cubic

8. To find the fluent of x × √a2 - x2.

9. To find a number from which if you take its square, there shall remain the greatest difference possible.

10. To rectify the arc DB of the circle DBRS. [A figure in the margin shews that an arc of any length is meant.]

The problem paper for 1786 is as follows.

1. To determine the velocity with which a Body must be thrown, in a direction parallel to the Horizon, so as to become a secondary planet to the Earth; as also to describe a parabola, and never return.

D2'

2. To demonstrate, supposing the force to vary as how far a body must fall both within and without the Circle to acquire the Velocity with which a body revolves in a Circle.

3. Suppose a body to be turned (sic) upwards with the Velocity with which it revolves in an Ellipse, how high will it ascend? The same is asked supposing it to move in a parabola.

4. Suppose a force varying first as

1

D3,

secondly in a greater ratio

of these Cases to determine whether at all, and where the body parting from the higher Apsid will come to the lower.

5. To determine in what situation of the moon's Apsids they go most forwards, and in what situation of her Nodes the Nodes go most backwards, and why?

6. In the cubic equation x3+qx+r=0 which wants the second term; supposing x=a+b and 3ab= q, to determine the value of x.

9. To find the fluxion of the mth power of the Logarithm of x.

10. Of right-angled Triangles containing a given Area to find that whereof the sum of the two legs AB+BC shall be the least possible. [This and the two following questions are illustrated by diagrams. The angle at B is the right angle.]

11. To find the Surface of the Cone ABC. [The cone is a right one on a circular base.]

12. To rectify the arc DB of the semicircle DBV.

I insert here the following letter from William Gooch, of Caius, in which he describes his examination in the senatehouse in 1791. It must be remembered that it is the letter of an undergraduate addressed to his father and mother, and was not intended either for preservation or publication—a fact which certainly does not detract from its value. His account of his acts in 1790 was printed in the last chapter. This letter is dated January, 1791, and is written almost like a diary.

'Monday aft. 12.

We have been examin'd this Morning in pure Mathematics & I've hitherto kept just about even with Peacock which is much more than I expected. We are going at 1 o'clock to be examin'd till 3 in Philosophy.

From 1 till 7 I did more than Peacock; But who did most at Moderator's Rooms this Evening from 7 till 9, I don't know yet;-but I did above three times as much as the Sen' Wrangler last year, yet I'm afraid not so much as Peacock.

Between One & three o'Clock I wrote up 9 sheets of Scribbling Paper so you may suppose I was pretty fully employ'd.

Tuesday Night.

I've been shamefully us'd by Lax to-day;-Tho' his anxiety for Peacock must (of course) be very great, I never suspected that his Par

tially (sic) wd get the better of his Justice. I had entertain'd too high an opinion of him to suppose it.—he gave Peacock a long private Examination & then came to me (I hop'd) on the same subject, but 'twas only to Bully me as much as he could,-whatever I said (tho' right) he tried to convert into Nonsense by seeming to misunderstand me. However I don't entirely dispair of being first, tho' you see Lax seems determin'd that I shall not.—I had no Idea (before I went into the Senate-House) of being able to contend at all with Peacock.

Wednesday evening.

Peacock & I are still in perfect Equilibrio & the Examiners themselves can give no guess yet who is likely to be first;-a New Examiner (Wood of St. John's, who is reckon'd the first Mathematician in the University, for Waring doesn't reside) was call'd solely to examine Peacock & me only. but by this new Plan nothing is yet determin'd. So Wood is to examine us again to-morrow morning.

Thursday evening.

Peacock is declar'd first & I second,-Smith of this Coll. is either 8th or 9th & Lucas is either 10th or 11th.-Poor Quiz Carver is one of the oi Tо\\o¡ ;—I'm perfectly satisfied that the Senior Wranglership is Peacock's due, but certainly not so very indisputably as Lax pleases to represent it -I understand that he asserts 'twas 5 to 4 in Peacock's favor. Now Peacock & I have explain'd to each other how we went on, & can prove indisputably that it wasn't 20 to 19 in his favor;—I cannot therefore be displeas'd for being plac'd second, tho' I'm provov'd (sic) with Lax for his false report (so much beneath the Character of a Gentleman.)—

N.B. it is my very particular Request that you don't mention Lax's behaviour to me to any one1.'

It was about this time that the custom of printing the problem (but not the other) papers was introduced.

Such was the form ultimately taken by the senate-house examination, a form which it substantially retained without alteration for nearly half a century, and which may fairly be considered as the archetype of the numerous competitive examinations now existing in England. It soon became the sole test by which candidates were judged. In 1790 James Blackburn of Trinity, a questionist of exceptional abilities, was informed that in spite of his good disputations he would not be allowed a degree unless he also satisfied the examiners

1 Scholae academicae, 322-23.