to demonstrate this proposition, but his reasoning appears to be inconclusive. It is as follows:

Any curve, or any polygonal line which envelops the convex line AMB from one end to the other, is longer than AMB, the enveloped line.

E

We have already said, that by the term convex line, we understand a line, polygonal, or curve, or partly curve and partly polygonal, such that a straight line cannot cut it in more than two points. If in the line AMB there were any sinuosities or re-entrant portions, it would cease to be convex, because a straight line might evidently cut it in more than two points. The arcs of a circle are essentially convex ; but the present proposition extends to any line.

which fulfils the required conditions.

Р

M

B

of

This being premised, if the line AMB is not shorter than any those which envelop it, there will be found among the latter a line, shorter than all the rest, which is shorter than AMB, or at most, equal to it. Let ACDEB be this enveloping line: any where between these two lines draw the straight line PQ, not meeting, or at least only touching, the line AMB. The straight line PQ is shorter than PCDEQ; hence, if instead of the part PCDEQ, we substitute the straight line PQ, the enveloping line APQB will be shorter than APDQB. But, by hypothesis, this latter was shorter than any other; hence, that hypothesis was false; hence all the enveloping lines are longer than AMB.

Now all that this reasoning proves is, that it is impossible to find, among the enveloping lines, a line shorter than all the rest; for whatever line be supposed the shortest, one shorter still may always be found. If indeed such a line could be found, then, if the hypothesis that this line is shorter than, or equal to, AMB, could be shown to be impossible, the truth of the theorem would be indisputably established. Legendre has doubtless deceived himself in the foregoing reasoning; he has unconsciously set out with two hypotheses, and having shown one to be impossible, infers, unwarrantably, the impossibility of the other.

A more satisfactory method of obtaining this conclusion is given by

M. Develey, in the notes to his Elémens de Géométrie, the substance of which I shall here give.

Take either of the lines that can envelop the convex line AGHB, the line ACEDB for example; there will necessarily be some space between these two lines, otherwise they would be identical, and would, in reality, form but one. Through a point in

A

E

D

G

H

n

B

this space draw a straight line, which may meet the enveloping line in two points m, n, but not cut the enveloped line. Then we shall have mn ▲ mEDn, and consequently ACmnB ACmEnB. This proves that whatever enveloping line is taken, a shorter can always

be found.

By repeating this construction, and proceeding from one enveloping line to a second and shorter line, from this to a third still shorter, and so on, we shall observe that the spaces inclosed by the enveloping lines become evidently smaller and smaller, and consequently always approach to the space contained by the enveloped line; that is, to the space AGHBA, which is doubtless smaller than either of the former. All these effects continue as long as there is any space between the enveloping line and the other. It appears, therefore, that since these enveloping lines decrease in length, as the space which separates them from the enveloped line diminishes, if there could be a last enveloping line it would in reality coincide with that enveloped; this therefore is shorter than all the rest.

BOOK VIII.

This book, like the fourth, is entirely practical: the construction of the fifth problem is taken from Mr. Leslie's Geometry. The last fifteen problems relate to the division of surfaces, an important part of practical geometry, although seldom noticed in the elements. The geometrical constructions of these problems I have framed so as to suggest the most convenient analytical solutions, in order that they may the more readily be applied in matters of real practice, such as

the division of fields, &c. It does not appear necessary to exhibit here the analytical expression deducible from each construction, as they may be very easily inferred by the student.

On Propositions XXXIII, and XXXIV.

The area of the space included between two concentric circles, or of a circular ring, as it is sometimes called, is readily determined from knowing the diameters of the concentric circles.

Thus, if AB, CD be the respective diameters of two concentric circles, of which O is the common centre, then calling the surface of the inner circle s and that of the outer S, while # is put for the number 3.1416, we have, by the scholium to proposition XIII. of Book VII., the following expression for the surface of the ring, viz.,

Now

S — s = π (OB2 — OD2)

ALC

E

D

OB2 — OD2 = (OB + OD)·(OB—OD)=AD·DB; also AD DB is equal to the square of the perpendicular DE, which is a tangent to the inner circle at D, consequently

S-ST.DE';

that is, the surface of the ring is equivalent to that of a circle, whose radius is equal to the tangent DE.

The two problems in the text were first solved by the late venerable and ingenious Dr. Hutton, who appears to have set high value upon these solutions: the first solution was pirated very shortly after its publication by a Mr. Clark, whose conduct in this affair Dr. Hutton very indignantly reprobates, in the third volume of his valuable Mathematical Tracts; where an account of the origin of these problems is given at length, particularly of the last, which is by far the most curious and difficult of the two. I cannot resist the desire which I feel to gratify the student, by giving Dr. Hutton's account of this problem in his own interesting manner, with which extract I shall terminate these notes. After discussing the former problem, he proceeds thus:

"With respect to the other curious and kindred problem, that of dividing a given circle into any number of parts that may be mutually equal both in area and perimeter, some account of its rise has been already given. It was first anonymously proposed in the year 1774 as a curious paradoxical problem, but unaccompanied by the least hint or intimation of any mode of solution whatever. It was indeed, announced by the proposer expressly as a seeming paradox, but accompanied by the declaration, that it nevertheless was capable of a strict geometrical solution. The problem remained however some time unanswered, being given up by all persons as a matter quite hopeless; and by most deemed, in fact, as little to be expected as the quadrature of the circle itself, to which it was thought to be nearly allied, and indeed dependent on it; for no person could imagine any other possible way of a circle being divided, even in idea, into any number of such parts that might be equal, both in area and perimeter, than by radii drawn from the centre to the point of equal divisions in the circumference: this was, in effect, reducing the problem to this other, of dividing the circumference in any proposed number of equal parts, which was deemed on all hands a thing impossible to be effected. After some time no person thought any more of the matter, but as a thing never to be accomplished; and so I believe it might have remained to this day, but for the occurrence of some such accident as that which actually led myself into the train of thought which soon ended in the complete solution. The construction I first inserted in the Critical Review ; next it was introduced into my first, or quarto volume of Tracts, published in the year 1786, accompanied with a short account of its rise, and a considerable improvement of it, by rendering the property general for the division into all ratios of parts, equal or unequal, and extending the same to all ellipses, as well as circles. After which, I have usually been in the habit of introducing it into my Dictionary, and the more common elementary books on mensuration," &c.

END OF THE NOTES ON THE FIRST EIGHT BOOKS.