This Letter must be taken-as it was intended to be taken-ironically; and the last paragraph is positively untrue. In a Paper read at a Literary and Philosophical Society I stated, and had Captain Judkin's permission to do so, that between the latitudes of 40° and 50°, he could never make his "ship's place," as found by a lunar observation, agree within 10 to 15 miles with his ship's position as found by chronometer, even under the most favourable circumstances for taking his lunar observation: and on no other occasion have I ever made a public use of Captain Judkin's name.

Now, Bennett's construction of Fig. 2 is perfectly correct. Future Geometers may give "the honour and glory" of discovering the geometrical quadrature to Bennett, (be it so), which according to Mr. J. Radford Young is of "no theoretical or practical value." What matters it, who gets "the honour and glory"? I have my reward in the consciousness of having used my best endeavours, to deliver geometrical and mathematical science from the absurdities that have hitherto shrouded them with reference to 's true value.

Well, then, would it not be very absurd if I were to say that either Bennett or myself constructed the figure by the "rule of thumb"? Would it not be equally absurd if I were to say that I can furnish the proof that the square and circle are equal in superficial area, without the aid of numbers? Mathematicians have never admitted Bennett's construction of the figure, and if they admit the construction to be correct, how can they prove our statement that the square and circle are equal in superficial area, to be untrue, either with or without the aid of numbers? It is self-evident that AD and BC are

diameters of the circle, by construction; and it follows, that O A, OB, O D and OC are radii of the circle; and it is admitted that "we may make radius unity:" but, even if this be not admitted, it cannot be denied, that Mathematicians make radius unity their starting point, in the search after the circumference of a circle of diameter unity; or in other words, in their search for the ratio of diameter to circumference in a circle: and they assume the symbol to denote the circumference of a circle of diameter unity. The higher branches of mathematics have done nothing towards finding 's true value, and so far as our "recognised Mathematicians" are concerned, "still lies lurking in his den." Mathematicians may assert that a square equal in surface to a given circle, cannot be constructed, "isolated and exhibited," but this is a mere assertion without a shadow of proof, and, moreover, is not true. Well, then, there is one point upon which all Mathematicians must be agreed, and that is, that if A O, a radius of the circle in Fig 2, = unity, the length of B A C a semi-circumference of the circle will be equal to the arithmetical value of the symbol π.

The geometrical figure represented by the diagram (Fig. 2), is constructed as follows:-Draw two straight lines at right angles, intersecting at the point O, and with O as centre, and any radius, describe the circle. It is self-evident, that A D and B C are diameters of the circle. Produce O A, O B, O D, and OC to the points E, F, G, and H, making A E, BF, DG, and OH equal to one fourth part of the radius of the circle, and join E F, F G, GH, and H E, and so construct the square EFG H, which is equal in superficial area to the circle.

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Proof: Making radius unity, OA a radius of the circle = 1. (O A + 1 0 A) = (1 + ·25) = 1·25 (OA+OA) O E, and OF, OG, and OH = O E, by construction; and it follows, that 2 (O E) = 2(1′25) 2'5 E G, a diagonal of the square E F G H. But, by Euclid, Prop. 47: Book 1: (O E2 + O F2) = (1′252 + 1°25') (1.5625+ 15625) 3125 EF area of the square E F G H ; therefore, √EF= √3125 = EF, a side of the square EFGH. But, 3'125 (OA2)=(3′125 × 1 × 1) = 3·125= area of the square EFGH; and since area in every

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circle, it follows, that (O A2) = area of the circle. But, FG EF, by construction; and by Euclid: Prop. 47: Book I.: E F + FG (3125+3125) = 625 √6:25 = 25 E G.

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= (√3·125 + √3·1252) = EG; therefore, VEG

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Hence: The ratio between the diameter of the circle and diagonal of the square EFGH is as 2 to 25; and when Mathematicians can prove that this is not the true ratio between A D and E G, they will be able to prove that the square E F G H and the circle are not equal in superficial area-but not till then.

The geometrical figure represented by the diagram, Fig. 3, appears at first sight to be somewhat complicated, but on careful examination it will be found to be extremely simple. It is readly constructed in the following way :—

Draw two straight lines at right angles intersecting at O. With O as centre and any radius, describe the circle X, and about it circumscribe the square E F G H. With each of the angles of the square, E F G H as centre, and a demi-semi-radius of the circle X as interval, describe the four small circles. Join a b, cd, ep, mn, n b, bd, dp, pn, ac, ce, em, ma, AC, CB, BD, and D A.

This remarkable geometrical figure is a study for Geometers, and I shall not go into all its properties; indeed, if I were to attempt to do so, I should have to go over much of the same ground that I have already "trodden" in my Letters of the 9th, and 16th November, 1868, to Mr. J. M. Wilson, Mathematical master of Rugby School, and those Letters are in print: but I may, without impropriety, direct the attention of Mathematicians to certain facts, which I have not brought out in those Letters.

The eight angles A O a, A On, DOM, DOp, BO e, BOd, CO c, and C O b, at the centre of the circle X, are equal angles of 36° 52': and the four angles a O b, c O d, e Op, and m On, at the centre of the circle are equal angles of 16° 16'; and the twelve angles are together equal to four right angles; and having fixed 90o as the measure of a right angle = 360°. No good Geometer, even if his knowledge of Mathematics be somewhat limited-if the knowledge he has be rightly applied-can have the slightest difficulty in convincing himself of the numerical values of these angles. Now, if we take one of the squares on the radius of the circle X, say the square. OBF C, and draw the diagonal O F, bisecting the angle O and its subtending chord cd at a point, say N, dividing the isosceles triangle Ocd into two equal right-angled triangles ONc and ONd; it follows, that the angles NOC and NO d, will be angles of 8° 8': but no man in the world can prove this without the aid of mathematics: not only so, but no man can prove the value of the angles at the centre of the circle, unless he know how to make a right use of mathematics. Had Mr. Gibbons known how to rightly apply Mathematics to Geometry, we should not have found him making the angles COC and BOd angles of 36° 52' +x, and the angles NO and NO d angles of 8° 8' — y.

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Again: OB is a radius of the circle X: 3 (0 B) = Bd Be, by construction: and (BD) or (Be) = Ge = Fd, by construction. The sum of O B and Be Fe: and similarly, the sum of O B and B d = Gd: the difference of OB and Be Ge, and similarly, the difference of O B and B d= Fd. But, Fc Fd, by construction, and F is a right angle; therefore, C Fe is a

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