is given by way of illustration, and the enunciations of several others are added as exercises for the student.

QUESTION VIII.

If perpendiculars be let fall from the angular points of a triangle upon the sides; and if, from the extremity of one of these perpendiculars, lines be drawn to the other two perpendiculars, and also to the two remaining sides, making equal angles with them all on the same side, then the feet of these lines shall all range in the same straight line.

Solution. Let A B C (fig. 8), be the triangle, A E, B F, CD, the perpendiculars meeting each other in O, and the sides in E, D, F, respectively. From the point D, draw D P, D Q, DR, D S, making equal angles with B C, B F, A E, and A C. Then since D is a point in the circle circumscribing the triangle B C F, the points P, Q, S, are in a right line. (See Diary, 1845, p, 62). And since D is also a point in the circle circumscribing the triangle AFO; the points R, S, Q, are also in a right line. Hence the points P, Q, R, S, are in a right line, as stated in the enunciation. Q. E. D. Theorem I. Let A B C be any plane triangle, A E, CD, B F, the perpendiculars meeting the sides in E, D, F, respectively, and let L be their point of intersection. Bisect A C, B C, and A B, in P, Q, R, and erect the perpendiculars P G, Q H, and R I, respectively equal to LB, LA, LC. Join I G, GH, HI, and the triangle A B C IGH.

Theorem II. The continued product of the radii of all the circles which can be described each to pass through the angular points of any plane triangle, and cut the circumscribing circle and a circle of contact at right angles a b c (a + b + c) ▲ 12 16 (a-b) (b-c) (c-a).

=

Theorem III. Let three parabolas be described to the three sides of a plane triangle, and have their principal axes in the lines bisecting the angles, and let a normal be drawn from

the point of contact of each parabola with that side which its principal axis intersects; then the product of these normals is equal to the product of the distances of the angular points of the triangle from the centre of the inscribed circle.

Did space permit, I might add considerably in illustration of other portions of the ancient geometry, and give his generalizations of some of the most curious questions in the "Mathematical Companion." Mr. Butterworth was always fond of such extensions, and no doubt would urge his pupils to make similar attempts. Some of Mr. Buckley's latest papers contain a series of interesting problems of this nature; but the preceding will suffice to prove that he was a geometer of no mean order.

In almost every instance the Greek models are faithfully copied, and in some instances the variations of the data, with the corresponding modifications in the constructions, are pointed out with a minuteness worthy of the Alexandrian school.

In his earlier papers the historical bearings of the questions under consideration are not very well defined. This is not the case with some of his later discussions, several of which are very carefully annotated. This can be satisfactorily accounted for from the fact that in early life he was unable to procure the necessary works of reference; but, on attaining middle age, his abilities and high moral character had become known; he was soon engaged in several important offices of trust, and was fortunate enough to marry a lady who was possessed of a genial disposition, and considerable property. From this period his library began to increase. Every work of note was purchased as soon as published; and besides these he collected a large number of the works of the continental geometers, both ancient and modern; so that at the time of his decease he had formed one of the most curious and valuable collections in the district.

But the possession of property did not prove to him an unmixed good. He became involved in a long and expensive Chancery suit respecting some portion of a woollen mill at Delph; and although he ultimately sustained his rights, it was not without considerable expense, both of time and money. During one portion of the proceedings he was obliged to retain possession of the property by sleeping on the premises. The room was cold, damp and unhealthy; and he unfortunately contracted a disease which developed into consumption. After suffering for a considerable period, his naturally robust constitution gave way, and he died as a Christian should. This event happened on the 15th of July, 1856, when he was in the 47th year of his age: his widow followed him to the grave exactly three years after, and their ashes rest together in the burial ground attached to St. Thomas's Church, Friarmere, under a marble slab on which an inscription appropriately commemorates both the virtues and the abilities of the deceased.

ON THE

SCARCITY OF HOME-GROWN FRUITS IN GREAT BRITAIN; WITH REMEDIAL SUGGESTIONS.

By C. Roach Smith, F.S.A. &c.

[In a Letter to Joseph Mayer, F.S.A. &c., Vice-President.]

MY DEAR SIR,

(READ 7TH MAY, 1863.)

Since I have resided in the country I have been led to consider a fact which does not appear to have excited the attention of the public, although it is one of great importance, affecting the health and comfort of the great majority of the population; indeed, it may be said, of almost the whole of the working classes and particularly of those residing in towns. It is the great scarcity and consequent high price of fruit. I have been at some pains to ascertain the nature of the food which supplies the daily wants of the cottager, of the townsman, of our soldiers and of our sailors; and I am well convinced that, in a sanitary point of view, the day is not far distant when the Government will feel bound to make inquiry and help to remedy an evil the effects of which may be most disastrous.

Nature intended that fruit and vegetables should constitute at least the chief support of man. They are indispensable to his healthful existence, as may be proved and is yearly strikingly proved, by depriving ships' crews of this species of food. Some years since the boys in Christ's Hospital were so infected with cutaneous diseases that they were compelled to be sent home; and so general and bad was the malady that a medical

I

inquiry was instituted. The evil was proved to have resulted from the want of fruit and vegetable diet. Now if a school such as this, richly endowed and watched over, is liable to be infected with loathsome disease from the absence of natural food, can we be surprised that millions of our fellow-creatures, steeped in poverty and ignorance and of improvident habits, are the victims of a neglect or misunderstanding of one of nature's primary laws ?

That my remarks may not take a discursive range, or seem indefinite, I will limit them to one or two of the main fruits which should be upon every poor man's table; but which have become so scarce and expensive that they are almost luxuries upon the tables of the rich. First of all stands The Apple. We learn from Solinus that Apples were, in his time, so common in Thule* (Shetland) as to constitute the chief winter food of the people. But, at the present day, how many of the Irish have ever seen an Apple-pudding? Even in Kent (a reputed fruit county), it is nothing uncommon to find Apples towards the month of January fetching from 28. to 3s. a gallon; and I have known them supplied to a nobleman's family in the spring, at the rate of from 4d. to 6d. per Apple! Now the climate of Ireland is not worse than it was in the time of the Romans. Britain seemed then, much as we find it, always covered with fogs; but Apples grew abundantly; and would flourish much more plentifully now, because a wide extent of waste land has been brought into a state of cultivation; and we will admit there is more horticultural science.

Am I attaching too much importance to the Apple ? Produce it upon the dinner table in any shape, and nine hundred

* Thyle larga est, et diutina, pomorum copiosa. Qui illic habitant, principio veris inter pecudes pabulis vivunt, et lacte in hyeme compergunt arborum fructibus. Cap. 35, edit. Ven. 1473. See Mr. Hogg's remarks on this passage in his Paper on Iceland, printed in The Transactions of the Royal Society of Literature," vol. vi, p. 324 et seq. The "ultima Thule" of the Romans was probably Mainland, the chief of the Shetland Islands.