Sir,-In your Magazine of the 6th Oct., I perceive there is an attempt made by Captain Fitzmaurice to claim the first idea of Galloway's Rotary Engine, but now styled the Orkney Rotary Engine.

Now, Sir, I think it only justice to give the merit (if any) of the invention where it is due.

Having been employed by Mr. Galloway to superintend his business, and make working and other drawings for him, I had an opportunity of knowing what his ideas were almost as soon as they were formed, which was very rapidly. On attending at the office one morning, as usual, Mr. G. showed me a sketch which he had made when the thought

struck him he asked me to make a diagram of it, and also a section of the cylinder and piston in wood, which I did; and I can positively affirm that Captain Fitzmaurice knew nothing of it until it was in this tangible form. I was with Mr. Galloway when he introduced the invention to the notice of Captain Fitzmaurice, who was very much struck with it; and Mr. G. then made some pecuniary arrangements about including it in a patent he was then about taking for further improvements in his locomotive engine for ascending inclines.

We collect the following particulars from a return to an Order of the House of Commons, which has just been published:

1

1846

1847

- 1848

- 1849

Of the 219 placed on trial, 79 were afterwards purchased for the public use.

Hutton .(138) Frodsham.. (2074) Hewitt (1177) Eiffe... (662)

....

......

The highest prices given were, 681. 58. for Hutton (138), and the like sum for

Frodsham (2074). Twenty-one obtained only 421., and a few much less. The largest number purchased from any one maker was 14, from Loseby, but owing, apparently, to other circumstances than the position which that gentleman held in the competition.

Mr. Loseby is the inventor of a chronometer compensation of great merit, and it was avowedly to reward him (in some measure) for that invention, that the Admiralty gave him such a preference in their orders. The defect in chronometers which it was the object of Mr. Loseby's invention to remove, and which it is admitted to have removed most successfully, was this-that if the compensation is perfectly adjusted for very high and very low temperatures, the chronometer gains at middle temperatures. The way in which Mr. Loseby rectifies this defect is, to attach to the balances of his chronometers curved tubes containing mercury. The mercury, on expanding with an increasing temperature, arrives in parts of the tubes inclined in different degrees to the radii of the balance, and therefore its successive expansions produce successive effects of different magnitude on the momentum of the inertia of the balance; and by giving different forms to the tubes containing the mercury, the law of the successive alterations of the momentum of inertia may be made to adapt itself to the law of alteration of the elasticity of the spring, whatever that law may be. The Astronomer Royal (Mr. Airy), in reporting to the Admiralty on this compensation (28th May, 1845), says, "I consider this contrivance (taking advantage very happily of the two distinguishing properties of mercury, its fluidity and its great thermal expansion), as the most ingenious I have seen, and the most perfectly adaptable to the wants of chronometers. I am not aware that it is liable to any special inconvenience." He was pleased at the same time to add, "No construction whatever for this purpose, however successful, can now, in my opinion, claim any pecuniary reward." And in a subsequent report (19th Feb., 1846), he gives this as his reason, "The nature of the defect, and of the modes of remedying it, were pointed out strongly and clearly by Eiffe, and contrivances for correcting it to a very considerable degree of exactness were actually adjusted

by him; and after this has been once done, the merit of arranging a new appa ratus for the same purpose, however ingenious (and Mr. Loseby's is really very ingenious), is very small." Mr. Airy, therefore, gave it as his opinion, that "the Admiralty should give encouragement to Loseby, not by giving him money (for a grant of which application had been made), but by applying to him for a few additional chronometers." The Admiralty followed the Astronomer Royal's recommendation; and in the opinion of another important functionary (the Hydrographer), they have done all that the circumstances of the case warranted.

"The statement of the Astronomer Royal," says the Hydrographer, Admiral Beaufort, "is perfectly correct. In his Report, May, 1845, he distinctly said, that no construction for the purpose that Mr. Loseby had in view ought to be pecuniarily rewarded; but, for very obvious reasons, their Lordships did not think it prudent to establish that as an inflexible rule, and much less to publish it.

"The immediate purpose of Mr. Loseby's construction was to resist great changes of temperature, in which he had been in some measure anticipated by Mr. Eiffe; and the agent that Mr. Loseby adopted, mercury, had been already applied by M. Le Roy; yet the means by which Mr. Loseby employed that agent were new and very ingenious.

"Ultimate success, however, could not be proved by short artificial trials at home, and therefore the Admiralty, though refusing him a direct reward, have afforded him, by spreading his chronometers through all climates, the best and most satisfactory means of establishing the merits of his invention.

"In doing this, they have carried out the Astronomer Royal's principle of general encouragement, and to a great extent, as they have purchased thirteen of Mr. Loseby's chronometers, and paid him for them 6307."

multiples of the sine, cosine, &c. By Mr. William Wilkins.

The following tribute to the memory of this gentleman appears in the Diary for 1778, and as it is perhaps the only existing record of him, we give it entire :

"The proposer (of Ques. 19) Mr. William Wilkins, intended to have given a solution, but untimely death prevented him; we mention with the utmost concern the loss of this very ingenious young gentleman, who, to a most engaging and amiable disposition, united those talents which bade fair to have rendered him one of the greatest mathematicians of the age. He died on January 15, 1777, in the 24th year of his age."

Art. XIV. A Specimen of a Method of Finding Rules for the Extraction of Roots. By Reuben Burrow.

This paper contains approximating rules for the square and cube roots of numbers, which are virtually the same as those to be found in most treatises on arithmetic. From Art. XX. it appears that Mr. Robertson, librarian to the Royal Society, was the first who gave the rule under the form to which Mr. Burrow's demonstration applies; but for a general investigation, extended to "any root whatever," reference may be made to Dr. Hutton's Tracts, vol. i., p. 213, or to Davies's Hutton, vol. i., p. 73.

Art. XV. Remarks on some Criticisms concerning the Property of the Lever. By Reuben Burrow.

The paper is intended as an answer to Dr. Hamilton's objections "against Sir Isaac Newton's proof of the Property of the Lever." The objections themselves may be seen in the first of his "Philosophical Essays," viz., that on "Mechanic Powers," which was reprinted in Part I. (and last) of the "Mathematical and Philosophical Tracts," published by Glendinning in

1801.

Art. XVI. An easy method of determining experimentally the Curve which a shall describe in its Flight. By Reuben Burrow.

Art. XVII. Additional Remarks on the Equation of Payments. By Reuben Burrow.

The Equation of Payments appears to have attracted considerable attention during the publication of this

periodical, for Articles I., XI., and XVII., are entirely devoted to its consideration; and it also forms the subject of several questions both in this work, the Old Ladies' Diary, its "Supplement,” and Whiting's Scientific Receptacle. In Art. I. of this work, Mr. Dalby offers an improvement of Malcolm's method, and deduces formulæ applicable to both simple and compound interest. Mr. Burrow takes up the discussion in Art. XI., and after deducing formulæ for both cases, remarks that since, "Mr. Professor Hutton, F.R.S., has thought proper to condemn Kersey's rule as false, and to give the preference to a rule of Mr. Malcolm's, which he says is the only true one,' it will not be improper here to show that Malcolm's and Kersey's are in effect the same, and that both agree with the foregoing rule, when compound interest is allowed." This he accordingly does, and after deducing the conclusion" that the common method of computing the equated time at simple interest is true, and that Kersey's rule is also true in compound interest;" winds up this paper by observing that Mr. Dalby had arrived at the same results independently, so that "Professor Hutton's assertions to the contrary, have just as much validity as Dr. Horsley's confirmation of Stewart's theory of the sun's distance; and the same answer which Mr. Lander gave the Doctor is equally applicable to the Professor." Mr. Todd next appears as the proposer of Ques. 22, deducing from

his solution that "Malcolm's method always gives more money to the creditor than could be made by receiving the debts as they become due." In the Diary for 1779, Mr. Burrow re-appears, apparently in self-defence, and introduces his "Additional Remarks" by stating that in consequence of "The ingenious and learned Professor Hutton, Esq., having in the last edition of his Arithmetic, introduced a new and very polite method of confuting the arguments advanced in the Diary for 1777, on the subject of equation of payments; viz., by representing the writer as a malicious defamer and an ignorant pretender;' and notwithstanding the authority of so considerable a personage, there being still many people so obstinate as to retain their former opinion, that abuse is not demonstration, and that false

the rest were either omitted from necessity, or were sufficiently evident from previous solutions.

Most of the questions appear to have been selected with considerable care, and in the earlier numbers of the work especially, were well adapted to test the varied abilities of the Editor's numerous and able correspondents. The geometrical portion of the work is well sustained, nor will its preponderance afford any reasons for regret, since it preserves to us many valuable researches of those geometers of the last century, who may truly be said to have caught the mantle of Simson. In several of the first numbers the intended Editor and his correspondents laid the foundation of several inquiries which have since been found very prolific in the hands of succeeding geometers both of our own and other countries. Mr. Dalby, under the signature of "Caput Mortuum," and also in his real name, furnished the work with a series of investigations of singular ability and elegance; nor must we omit to particularize the contributions of Lawson, Crackelt, John Burrow, Sanderson, Moss, and last, but not least, those of Jeremiah Ainsworth, the friend and Tutor of Wolfenden; all of whom enriched the earlier portion of this periodical with many valuable and original questions and solutions.

Ques. 11, gives "the line bisecting the base, the difference of the sides, and the difference of the angles at the base,' to determine the triangle. The question had previously been proposed by Mr. Thomas Hulme, as No. 24 in the "Mathematician," and was there solved algebraically by Mr. John Turner. It was here re-proposed by the Rev. W. Crackelt, who gave a geometrical construction to the problem, not only when the sum, but also when the difference of the sides is given.

Ques. 21, by Mr. Thomas Moss, is perhaps the most important question ever proposed in an English periodical, since it approaches more nearly to a formal statement of some of the leading properties of the complete quadrilateral than any other we have seen. A particular case of the general problem was proposed by Mr. R [ollinson] as the 14th Question in the Mathematician, and the property was afterwards generalized by Mr. Davies in Question 18 of Clay's Scientific

Receptacle; but the method of investigation there used, does not seem well adapted for evolving the many interesting properties of the figure, and is certainly much inferior to that used by Mr. John Burrow in his discussion of the question under consideration. Indeed, Mr. Burrow's investigation, except as to form, differs very little from the present methods of treating these subjects, and, independently of its intrinsic merits, deserves greater publicity than it has hitherto obtained, since it establishes, beyond a doubt, that to an almost unknown correspondent in an equally little known English periodical, we are indebted for, at least, the re-discovery of the fundamental theorems in one of the most interesting fields of modern research, and which has since been so effectively and extensively cultivated by Carnot, Chasles, and our own Professor Davies.

Question.

"If from the extremities, S and V, of the base of a triangle, STV, two lines be drawn through a given point, N, meeting TV and ST in C and A, and the line, TN be joined, meeting AC in B; also if from A and C parallel lines be drawn meeting the base in M and P, then will demonstration." AB: BC:: AM: CP;-required the

tice of the times, the preceding is enunThough in accordance with the pracciated merely as a property of the triandiagram (next page) that TASNVCT is gle, it is obvious from an inspection of the the complete quadrilateral whose three diagonals are AC, TN, and SV respectively, and hence any properties proved to be true for the case of the triangle, are equally true with respect to the corresponding parts of the more general figure. Now, in the earlier portion of Mr. Burrow's demonstration, it is shown that

AB: BC: AD: DC,

SQ: QV:: SD: DV;