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In the first place, if = c, there are only three values of x, which are

0, ±√(2bc + c3).

For, in this case, EF = Cd, and CE=0: the points E' and e' corresponding to (2bc+c) and (2bc+c3) respectively, being exterior to the circle.

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than greater c, or if EF be less

than Cd, then there are four values of x, two positive, and two

negative: two of these values, which are expressed by

c2 2

C

± √ √ { a2 + − c √/ (a2 + b2 + ") },

are less than a, and therefore correspond to points E and e within the circle and two, which are expressed by

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and which are greater than a, correspond to points E' and e', which are exterior to the circle.

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b

In the third place, if be less than c, or if EF be greater

than Cd, then there are no points E and e within the circle which fulfil the conditions of the problem, and two of values of x, expressed by

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become imaginary: of the two others, expressed by

c2

* √ √ { a2 + C + c √√/ (a2 + b + c )},

one is positive and the other negative, corresponding to points E' and e', which are exterior to the circle.

In this solution, the positive and negative roots admit of their primary and appropriate interpretation, being referred to a fixed point in a given straight line, and being estimated therefore in opposite directions from it: but the occurrence of imaginary roots intimates that there are no points E and e within the circle through which DE may be drawn so that EF may be of the

If the as

made in

of a problem be

arithmeti

tinue so

required magnitude: and it may be further observed, that there is no modified construction which the geometrical interpretation of such imaginary roots would suggest, which is compatible with the conditions of the problem*.

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1045. If the assumptions which are made with respect to sumptions the representation of lines or other magnitudes be arithmetical, one part of as opposed to symbolical, in one part of the solution of a problem, the solution the results which are obtained may be either erroneous or inconsistent, if the same principles of representation be not applied cal, they throughout. Thus, supposing a given chord must con- DE passes through a given point C of the throughout. diameter AB of a circle, and it is required to determine the segments CD and CE into which it is divided in C; then if we should denote AC by a and CB by b, representing lines drawn in opposite directions from C by symbols with the same signs, then the same principle of representation and interpretation would be found, in the result, to extend to the lines CD and CE, which are also drawn in opposite directions from C: for if we denote CD by x, and DE by c, we should find

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x (c − x) = ab,

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A

B

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D

where the values of x are both of them positive and express the segments CD and CE of the chord: and as the assumptions made respecting them are those of Arithmetical Algebra, the results

The problem in the text, which is one of more than common instruction and interest, has been discussed, though very imperfectly, by Carnot in his Geometrie de Position, by whom its results are appealed to as subversive of the ordinary theory of the interpretation of positive and negative symbols, when representing lines: he appears to have laboured under the impression, in common with D'Alembert, and others who preceded him, that such interpretations of signs were absolute and not relative, and that the opposition of direction which they symbolized, was to be sought for, not in a different but in the same straight line; thus CE and Ce, in the figure in the text, would be legitimately symbolized by + x and r; but if DE was denoted by +1, the line which was symbolized by - must be sought for in the production of ED, and not in a different line DE'.

This example is produced by D'Alembert, in his Dissertation sur les Quantitès Negatives, in the eighth volume of his Opuscules, as presenting an anomaly to the ordinary theory of interpreting negative quantiles.

may be properly considered to belong to the same science and

to respect quantity only and not direction.

1046. If however the point A, through which the given A problem

chord ED passes, be exterior to the circle, and if AC and AB, estimated in the same direction from A, were represented by a and b, an assumption consistent with the principles both of Symbolical and Arithmetical Algebra, and if we further assume AD = x and DE = c, we should arrive at the equation

and, therefore, obtain

x (c+x) = ab,

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A

C

E

one value of which, representing AD is positive, and the other representing EA is negative, and exclusively belongs therefore to Symbolical Algebra: for if AD and DE estimated in the same direction from A be represented by positive symbols x and c, then EA, which is formed by the production of ED in a direction opposite to AD and DE, must, in conformity with the principles of that science, be affected with a negative sign.

If, however, the same problem be solved by the principles of Arithmetical Algebra, we may represent AD by x, AE by y, AC by a, AB by b, and DE by c, and form the two equations

y-x=c,
xy = ab.

If we replace y by c+x, the second equation becomes x (c + x) = ab,

whose unique arithmetical root is

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if we farther replace this value of x in the first equation, we get

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which is likewise positive and arithmetical: the lines AD and AE, as well as the lines AC and AB are represented in this solution by positive symbols.

of the same class solved both symbolically

and arith

metically.

problems in

which Symbolical Algebra results

attainable

by Arith

A large 1047. In the largest class of examples, however, the equaclass of geometrical tions which are formed have exclusive reference to the magniand other tudes of the quantities which the symbols represent, and not to their affections, or, as in the case of lines, to their positions with respect to each other: in such cases, Symbolical Algebra will affords no give no results which are not generally obtainable by Arithwhich are metical Algebra, inasmuch as the introduction of expressions not equally affected with negative or imaginary signs must be considered as referrible to the general rules of Symbolical Algebra only, and not amenable therefore to the principles of interpretation which are found to be applicable under other circumstances. Thus, if we take the diameter AB of a circle, drawing Dcd perpendicular to it, and joining AD and Ad, and if we represent the magnitude of the diameter AB by 2r, the magnitude of the segment (or sagitta) AC by x, we shall find

metical Algebra.

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Ambiguities from roots of solution.

In this result, √2ra represents indifferently the magnitude of the chord AD or Ad, but has no reference whatever to the position of one or the other of them with respect to AE: the introduction of the negative result therefore, is entirely due to the rules of Symbolical Algebra, and is, in no respect, relative to the problem proposed: and though it is quite true that if AD be assumed to be represented in magnitude and in direction by √271, then AD' which is equal and opposite to AD (and also any other equal line which is parallel to it) may be correctly represented by√2ra, yet such an interpretation of the negative root, in the solution of the problem under consideration, would be a mere surplusage and altogether alien to the original assumptions which were made in other words 2rx may equally represent, under such circumstances, any line whatever which is equal to AD.

1048. The most common, however, of all the sources of ambiguity in the symbolical solution of problems, is due to the

This example is also discussed by D'Alembert in the same Dissertation, referred to before.

introduction of extraneous factors which are required for the rationalization of the equations of condition, which the problem furnishes, and the consequent introduction of roots which will satisfy the rationalized but not the primitive equation; we have already illustrated this theory of proper roots of equations, as distinguished from roots of solution, in great detail, in a former Chapter (xx). The following is an example of a problem, whose solution leads to an equation of condition, of the peculiar class to which we are now referring.

"To determine a right-angled triangle the sum of whose Problem. two sides is 35, and of which the perpendicular let fall from the right angle upon the hypothenuse is equal to 12.”

If we denote one of the two sides by x, the conditions of the problem will give us the equation

35x − x3 = 12 √(2x2 − 70x + 1225),

which rationalized becomes

x* – 70x3 + 937 x2 + 10080x - 176400 = 0.

This biquadratic equation is resolvible, by the processes given in Chapter xx, into the two quadratic equations

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The roots of the first equation are 15 and 20, which satisfy the primitive equation of condition, and will be found to solve the problem: they express the two sides of the right-angled triangle.

The roots of the second equation, which are 47.4 and – 12.4 nearly, satisfy the rationalizing factor

35x-x2 + 12 √(2x2 − 70x + 1225) = 0

only, and not the primitive equation, and are therefore roots of solution only, and are not relative to the problem proposed.

1049. We must beware, however, lest we conclude too The roots

of a ra

hastily, in all cases, where an irrational equation is rationalized, tionalizing

sometimes

that some of the roots of the resulting equation are roots of solution factor are merely, and not relative to the problem proposed: for it may proper happen that all the irrational factors of the rationalized equation roots.

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