unit is the QUOTIENT of a number divided by itself, while the real unit is the product of a number multiplied by the inverse of that number. Let n be any number, n divided by n is a speculative unit, and n × is a real unit, as a rectangle would be. If, then, be the supposed unit, it is necessary, in order to obtain a real unit, to multiply by n; consequently, the nearer we approach one side, the farther we recede on the other.

n

The following solution, which for the first time is given of Pell's problem, is the only one exempt from this absurdity:

To find a, b, c, (See Wallis's Algebra, chap. 60, 62, &c.) let there be a series of concentric circles, (Plate XXVII. fig. 1) 1 1 1, 1 1 1, 1 1 1....1 1 1....1 1 1.... 1 1 1, and let

them be so described that we have O 1=11 = 11 =

11=

2 3

c = the sector 1 1 1

Then if the areas of these sectors be substituted for a, b, c, in the equation (1), (2), (3), according to the following method, their differences will reduce these equations to identical ones.

Demonstration.

This demonstration is founded upon a remarkable property of the concentric circles of this figure. This property is, that the areas of each of the rings intercepted between two consecutive circumferences are equal to the area of the central circle. If we take the area of the central circle for an unit, the areas of each of the rings will be = 1.

To prove this, let ◇ 1 be the radius of the central circle, we may

easily perceive that the radii of the successive circles are the hypotheneuses of right angled triangles, whose sides are,

1. The radius of the preceding circle. 2. A constant tangent equal to the radius of the central circle. The series of radii will be then expressed by 0 1 (√1, √2, √3, √ 4, &c.)..........(7)

Now the circumferences of circles are constantly proportional to their radii: if, then, we designate by 2 the ratio of the circumference to its radius, the series of circumferences will be expressed by the progression (○ 1)2 × 2 π [√ 1, √ 2, √ 3, √ 4, &c.] .. (8)

But the areas of circles are as the squares of their radii; so that the series of circles will be expressed by the progression (O 1). 2

·(9)

[1, 2, 3, 4, &c.] In this series (by taking away the common factor (O 1)2, 2 π) 1

expresses the area of the central circle; 2, 3, 4, &c. express those of the successive circles. If we take the difference of each of the areas of these consecutive circles, we shall have the areas of the rings. Now these differences are constantly equal to 1; consequently the areas of the rings are = 1.

It follows from the above conclusions that if we reckon the central circle for the first ring, the series of rings will be expressed by the common urdinals, 1st, 2d, 3d, 4th, &c. ..... (10) while the series of circles are expressed by the absolute numbers 1, 2, 3, 4, &c. ...

...(11)

These ordinal numbers follow the direct or inverse order: when they follow the direct order, the areas of the rings are positive; when they follow the inverse order, they are negative. These areas are constantly = 1, and are represented by the equation +1 = e

-

(12)

These equations, which are fundamental, I thus demonstrate: in every system of logarithms the logs. are exponents, and these exponents are ordinal numbers, because they are the indexes of the terms of a geometrical progression whose first term is 1. Now from the principles demonstrated by Euler (Introductio in Analysin Infinitorum, Cap. VIII. No. 39,) we may prove the truth of the two following equations:

......

(14)

Log. (— 1) = ± 2 (k + 1) π √ − 1 ................ (15) √1 (See Lacaille's Leçons de Mathematiques, Nos. 833, 834, 835.) Let e be the base of the hyperbolic logarithm: we have 1 = log. e.. ..(16) Consequently log. ( + 1) = ± 2 k π √1 log. e ......(17)

.(18)

and log. (1) ± 2 (k + 1) π I log. e = √ From the above we easily derive the equations (12), (13). The following is an explanation of these equations: k can only be an

ordinal number, because it is the only variable factor of the exponent ± 2 km √ ]. Thus 2 k represents that circumference whose place is designated by k.

±√ is a sign of impossibility, because it expresses a quantity greater than a maximum, and less than a minimum; but the area of each concentric ring is out of that circle which serves as a nucleus, and it is the diameter of that circle which is a maximum. The diameter, then, of the exterior circle of this ring is greater than a maximum. The whole area of the ring which exceeds this diameter is then proved imaginary, which shows that the sign

√ belongs absolutely to it. It now remains to explain the sign. A ring contains two circumferences of circles; to wit, an external and an internal. Now 2 only expresses one; the sign causes it to express two; which I thus prove :

Let yy = a a -xx

...

π

..(19) be the equation to a circle: if we take the value of y we shall have

y =

xx

Naa ...(20) Here the double sign indicates two ordinates of an equal length drawn from any particular point of the diameter on each side of it. The positive ordinates, designated by +, extend only to half the circle; and the negative ordinates, designated by -, extend to the other half in order to obtain the ordinates which extend to the whole circle, we must unite the two signs, as in ±. Now when this sign is accompanied by 1, it does not mean + or —, butand; because the imaginary quantities always go in pairs, and they cannot be separated without an absurdity, as I will prove. Thus let 1 11 be the tangent to the central circle. This tangent

101

is the greatest ordinate which can be drawn to the exterior circle without entering into the central circle 1 11: its middle point is at

11 1

the same time the smallest of those which can be drawn in the interior circle, since it is reduced to this point, 1, in which the two

ordinates coincide, the two ordinates, 11 and 11 having then the

point 1, which is common to both, and are connected by that

0

point. Thus they form a continued right line, which is expressed by√1. If we refer this expression to the interior circumference, we have ± √1 = 0, which is not imaginary, because then it is the sign 1, which ought to be considered as 0. If, on the contrary, we refer it to the extreme circumference, we have ± √ 1, which only ceases to be imaginary at the two points 1 1, which coincide with this extreme

circumference. To apply this principle to the double sine of the expression ± 2 kπ √ — 1, let us divide into two equal parts the part 1 1 of the diameter 1 1, which is intersected between the two

circumferences of the ring which extends beyond the central circle; and through the point of division let us draw the dotted concentric circle; the circumference of the dotted circle will be an arithmetical mean proportional between the two extreme circumferences of the ring. If we take this dotted circumference as a line of abscissa, it is clear it will cut all the sections of the diameters intercepted between the extreme circumference into two equal parts. Each of these half parts will be equal ordinates drawn on each side of the circumference of the dotted circle, this circumference being taken as a line of abscissa, and the two extreme circumferences will be the curves described by means of these ordinates. As all these ordinates are imaginary, they have only two real points, which are their two extremes: one of these two extremes is a point in the dotted circle, and the other is a point in one of the circles already described: these three circles are then composed only of insulated points, the points of the dotted circle are double, and those of the circle described are simple. 2√ 1 expresses the sum of the ± points of the dotted circle; that is to say, +21 is the sum of the points of the exterior circle, and 2 √, the 1, sum of the points of the interior circie. Resuming all this explanation, we find±2 k√ 1 is the sign of the description of two concentric circles forming a ring by assuming for a line of abscissa a third concentric circle whose circumference is an arithmetical proportional mean between the circumferences to be described, the same as aa-xx is the sign of the description of a simple circumference by taking its diameter for a line of the abscissa.

√

-

π

This granted, in order to resolve the equations (1), (2), (3), I begin by multiplying their second member by the second member of equation (12), which gives me

*The idea of my giving one area for the root of another area, may perhaps be cavilled at; but when we consider that the root of the area of the square A B C D (fig. 2) can only be the area of a rectangle, such as A a Cc or Ac Bb, the explanation will appear clear.

Το that these are the true roots of equations (22), (23), prove (24), I substitute these roots for a, b, c; then take the differential*, considering the sign as the differential sign. These substitutions

The experimental quantities of equation (31) are reducible to

This kind of differential is the true and strict meaning of Lemma II. Sect. II. Book II. of Newton's Principia. (Momentum Genitæ, &c.) The manner in which Newton has demonstrated this lemma entirely refutes every passible objection.