them. He also dwelt upon the fact of the auroral light having been seen by himself and others, for some time, hanging between two points of land. In answer to a question from Dr. Collingwood, he stated that the dark bands were certainly not the result of mere contrast with the brighter parts.'

EIGHTH ORDINARY MEETING.

ROYAL INSTITUTION, 21st January, 1861.

The REV. H. H. HIGGINS, M.A., PRESIDENT, in the Chair.

The Rev. H. J. HINDLEY, M.A., and the Rev. C. D. GINSBURG, were elected members of the Society.

Mr. FABERT exhibited a remarkably fine specimen of red coral (Tubipora musica) from the Chinese seas, and several other objects of interest.

The Rev. J. ROBBERDS drew attention to Mr. P. H. Holland's method of sealing safety-lamps, so that they could not be opened without detection. The plan was similar to that of the metallic fastenings formerly applied to envelopes..

Dr. COLLINGWOOD exhibited some of the microscopic collecting envelopes referred to at the last meeting.

The PRESIDENT drew attention to the contemplated anniversary festival of the Liverpool Naturalists' Field Club, to be held in St. George's Hall, in April next, and read a prospectus of the proposed plan. He invited the assistance of ladies and gentlemen interested in the subject.

The following paper was then read :

CONTRIBUTIONS TO NAUTICAL SCIENCE.

BY THOMAS DOBSON, B.A.,CANTAB.

(Head Master of the School-frigate "Conway.")

WHEN Newton was engaged in investigating the nature of the great mechanical laws that govern the material universe, he is said to have made use chiefly of the method of fluxions, the most efficient instrument of mathematical investigation known at that time; but he took care to clothe the results of his researches in a comparatively simple geometrical dress.

The wisdom of such a course will be appreciated by persons engaged in instruction, who have to communicate to others the knowledge of necessary truths at which the philosopher has already arrived only by a patient and skilful marshalling of his mathematical symbols; and this communication is only practicable with young students when the subject can be presented to them in an elementary form. The mathematician of our day employs in discovery the Differential and Integral Calculus, justly styled by Dr. Whewell, in his Philosophy of the Inductive Sciences, "the principal weapon by which the splendid triumphs of modern mathematics have been achieved." But the philosopher seldom condescends now to imitate the master of philosophy by announcing his discoveries to the world in such a simple form as to put them on a level with the capacity and attainments of all who may have an interest in understanding them. His is the noble ambition of discovering hidden truth, and he leaves to others the humble, but still useful and honourable, office of devising the best means of diffusing a knowledge of his discoveries. My daily avocations have led me to become a labourer in this humble sphere.

Of all men, the sailor is most indebted to the mathematician, who has framed the rules which the sailor practices and relies upon; and computed the numerical data which the sailor takes from his Nautical Almanac; data which embody the practical results of mathematical problems of the very highest order of difficulty, and which have taxed the powers of the greatest mathematicians from Newton's time to our own.

Nautical science, then, having thus been constructed by help of the higher mathematics, offers an ample field for simplification; and that such a process is most desirable will be obvious when we reflect how essential a clear knowledge both of the principles and practice of nautical science is to that numerous and valuable body of men who are responsible for all the lives and property afloat. Such knowledge is more than ever indispensable in these days of steamships, clippers, and rapid passages, when a merchant captain must strain every nerve -and what is much worse, run every risk—in order to satisfy an exacting public, by making a passage in the shortest possible time. It is evident that the danger from an error in the reckoning of a dull sailing vessel is much less than in that of a long, sharp clipper; on the principle that the further you go on the wrong road, the more you go wrong.

The first subject to which I shall ask your attention this evening is a question relating to practical navigation, and may be enunciated thus:-

"The direction of the wind, and the course of the ship, being known, required the direction of the sails, so that the ship may make the most headway." This problem belongs to the difficult class of "maxima and minima," which are most successfully attacked by means of the Differential Calculus, and thus I first accomplished its solution. But, anxious to bring it within the reach of my pupils, I reconsidered it, and first succeeded in solving it by means of plane trigonometry; but at last was rewarded by discovering the simple geometrical

proof which follows. I am not aware that this problem has been published in any form; it is certainly not mentioned in any of the numerous English and foreign works on navigation that I have consulted.

Let A B be the direction of the wind, B C the direction in which the ship is progressing, and B P the projection of a sail on the plane of the deck. Draw A P perpendicular to BP, A C perpendicular to B C, and P D perpendicular to A C. Then, if A B represent the force of the wind, A P will be the effective part of it, acting perpendicular to the sail, and P B the non-effective part, acting parallel to the sail.

Now, A P is equivalent in magnitude and direction to the two component forces A D and D P, of which A D perpendicular to B C produces leeway only; and D P parallel to B C produces headway only.

Since A P B and A C B are right angles, the points P and C lie in the circumference of a circle of which AB is the diameter; and P D is the perpendicular from a point in the arc A C on its chord A D C; and will obviously be greatest when P coincides with Q, the middle point of the arc A C. Hence, in order to make the most headway, the ship's sails and yards should, as nearly as the shrouds, &c., will allow, lie

midway between the ship's course and the direction of the wind.

In some of the most important practical applications of nautical astronomy, where two altitudes of a heavenly body are taken at an interval of a few hours, during which the vessel has been proceeding on her course, it is necessary to reduce the first altitude to what it would have been if it had been measured at the place where the ship is when the second observation is made.

My second contribution to nautical science is a simple elementary investigation of the value of the correction to be applied to the first altitude to compensate for the "run" of the ship, as it is called. This value, of course, is well known, but my proof is well adapted for instruction, inasmuch as it has the advantage of placing clearly before the student the things which he is required to reason about, and is made to depend upon the rule for parallel sailing-the simplest case in spherical trigonometry. In this case, as in several others, I had the alternative of either inventing a simple intelligible proof, or of giving the rule to my pupils without demonstration, and resting on authority alone, a mode of proceeding altogether inconsistent with sound teaching.