right, and the demonstration of the first part is inapplicable to a right angle.

PROP. 31. B. III.

To the enunciation of this proposition are added in the Greek text the following words, and the angle of a greater segment is greater than a right angle; and the angle of a less segment is less than a right angle. I have omitted this part of the proposition for the reasons assigned in N. to Prop. 16. B. 3. nor can I believe that these parts of Prop. 16 and 31 were ever written by Euclid, for the principles on which the demonstration of the latter depends, are subversive of those by which the former is proved, as must be evident to any person considering the subject.

PROP. 33. B. III.

In the Greek text the construction of the second case is different, AB being bisected, and a perpendicular erected at the point of bisection; but by this construction the demonstration is rendered more prolix. It is evident from Ax. 12. that the lines AO and BO must meet, for each angle OAB and OBA is less than a right angle.

DEFINITIONS OF BOOK IV.

I have omitted two definitions which are prefixed to this book in the Greek text, those of a rectilineal figure inscribed in a rectilineal figure, and circumscribed about a rectilineal figure, because they are useless, as Euclid does not treat of such inscription or circumscription.

PROP. 3. B. IV.

The tangents ML and MN must meet LN, for, if AB and BC be drawn, the angles towards L and N are less than two right angles; and since the two angles at D together with those at F are equal to four right angles, and the internal angles EDF and EFD are less

than two right angles, EDG and EFH are greater than two right angles, therefore, if AKB and AKC, which are equal to EDG and EFH, be taken away from the four angles at K, it is evident that the lines KB and KC make an angle at the side of M; and therefore, if BC be drawn, the angles between it and the tangents at the side of M are less than two right angles, and therefore the tangents meet at that side.

PROP. 4 & 5. B. IV.

It is evident from Ax. 12. that the right lines bisecting the angles of the triangle in prop. 4. must meet; and from the same Axiom it appears that, if DE be drawn, the perpendiculars DF and EF in prop. 5. must also meet.

PROP. 6. B. IV.

Euclid demonstrates, that the inscribed figure is equilateral by prop. 4. B. 1. whence it is evident that the bases AB and AD of the triangle AEB and AED are equal, and so on of the rest: but the demonstration from the equality of the arches seems to me better.

PROP. 7. B. IV.

This proposition, as it now stands in the Greek copies, is corrupt; for in the demonstration are inserted the words, that FH is a parallelogram, and therefore that FG and KH, also FK and GH are equal, which are totally useless, as the demonstration proceeds to shew that FG and KH are each equal to AC, and that FK and GH are each equal to BD, whence it follows, that FG, GH, HK and KF are equal to one another. A similar error has crept into the eighth proposition, where it is said that GD, BK, GA, AH and AD are parallelograms, which is not necessary to the demonstration.

PROP. 11. B. IV.

If the division of an angle into any number of parts be conceded, and the construction of an isosceles triangle, in which the angles at the base may be any multiples of the vertical angle, any figure of an odd number of sides can be inscribed in a circle; and if the angles at the base be the sesquialteral multiples of the vertical angles, figures of an even number of sides may be thus inscribed. Subtract unity from the number of sides, divide the remainder by two, and construct an isosceles triangle, in which the angles at the base are to the vertical angle as the quotient to unity, and by this triangle the required figure can be inscribed. Ex. gr. if a heptagon were to be inscribed, the angle at the base ought to have to the vertical angle the ratio of three to one: if an octagon, as 31 to 1.

COR. 5. PROP. 15. B. IV.

Hence it is evident, that the side of an equilateral triangle is incommensurable with the diameter of the circumscribing circle, for it has to the diameter the ratio of the square root of three to two.

PROP. 16. B. IV.

B. 3.

The circumference of a circle, which by this proposition is divided into fifteen parts, can, by bisecting each of these parts (1), be divided into thirty parts, (1)Prop.50 and again into 60, 120, &c. And also, if each of the four parts into which a circle is divided by prop. 6, be bisected, a circle can be divided into 8, and again into 16, 32, &c. In like manner, by prop. 15, it can be divided into 3, 6, 12, &c.; and by prop. 11. into 5, 10, 20, &c. But a geometrical method of dividing a circle into any given number of parts has never yet been discovered.

A celebrated theorem of Proclus concerning ordinate polygons should not here be omitted: the theorem is, that only three ordinate polygons can be so placed at

a point as to make a continuous surface; for this purpose it is necessary that the angle of the polygon should be an aliquot part of four right angles, since the angles at any point are equal to four right angles; therefore, as the angle of an equilateral triangle is the sixth part of four right angles, the angle of a square the fourth part, and the angle of an hexagon the third part, it is evident, that six equilateral triangles, four squares, or three hexagons can be so placed at a point as to make a continuous surface. But the angle of a pentagon is greater than the fourth part, and less than the third part, of four right angles; and the angle of any other ordinate polygon is greater than the third part of four right angles, and therefore they cannot be so placed at a point as to form a continuous surface.

DEF. 4. B. V.

By this definition Euclid determines what magnitudes he acknowledges as homogeneous; in general, quantities are considered homogeneous, which he considers heterogeneous, ex. gr. a finite and an infinite line; the angle of contact and a rectilineal angle.

DEF. 5. B. V.

How proportional magnitudes ought to be defined, is still a subject of controversy among geometers. Euclid defines them thus: The first of four magnitudes is said to have the same ratio to the second, which the third has to the fourth, when any equi-multiples whatsoever of the first and third being taken, and any equi-multiples whatsoever of the second and fourth being taken, the equi-multiples of the first and third either together exceed, are equal to, or are less than, the equi-multiples of the second and fourth. This definition is liable to the objection, that there is not the least resemblance between it and the common notions of similitude or equality of ratios: it must be confessed indeed, that Euclid has demonstrated that magnitudes thus related retain the same properties, however they may be inverted or subjected to the various changes

made use of by geometers; but this is not sufficient, the connexion ought to be shewn between this definition and that relation of two ratios to which the name of equality is commonly given. Against this objection Barrow ably argues in his mathematical lectures, and after having overturned the various theories invented since the time of Euclid, endeavours to put a stop to the controversy, and repress every attempt towards forming a new theory, in the following words, Per has naturas (scil. magnitudinum proportionalium) nil aliud intelligi quam rei definite nomini, quatenus in usu communi versatur, respondentes conceptus aut significatus aliquos imperfectos & indistinctos in scientiis minimè respiciendos, ad quos proinde nullatenus exigendæ sunt definitiones, imo secludendis & eliminandis iis, ipsorumque loco substituendis rerum certis distinctis atque claris ideis, efformantur definitiones. If this be true, Euclid's definition is certainly the best, for undoubtedly it is wholly foreign from common use; but it is evident from the work itself, that Euclid never formed the definitions of his Elements by such a rule, and even Barrow acknowledges (Lect. 6. A. D. 1664.) Quod expediat a facilioribus magisque familiaribus passionibus argumentandi initium sumere, seu subjectum definire. Here it is confessed that if a more familiar definition than Euclid's can be given, it ought to be preferred, provided all the properties of proportionals which are necessary can be demonstrated from it: that the definition I have given is of this nature, will, I hope, be acknowledged: borrowed from the common idea of proportion, it is nevertheless accurate, and from it all the properties of proportionals, including even Euclid's definition, are demonstrated.

That the young student may more easily understand this definition, I annex some examples of equal and unequal ratios. Let the ratios of 6 to 42 and of 9 to 63 be compared together, and as often as a submultiple, 1, 2, 3, (or any other) of the first antecedent is contained in the first consequent, so often the equisubmultiple 1, 3, 4 (or any other) of the latter antecedent is contained in its consequent.

Likewise if the given ratios be that of 8 to 128,

Ꮓ