SOI 



ALGEBRA. 



ALGEBRA. 



202 



the two lines are in the same direction. And if all lines be on the 

 axis of length, all the four operations are those of single algebra ; while, 

 if they be all on the unit-side of the axis of length, there is nothing 

 but pure arithmetic ; except when an impossible subtraction obliges us 

 either to reject the process, or to enlarge the definition and pass to the 

 opposite side of the axis of length. 



16. Make a positive and negative direction on the axis of direction, 

 thus : Choose a direction of revolution in which a line revolving round 

 the origin, and setting out from the axis of length, shall be said to 

 revolve positively. Let the unit revolve positively, and let the first side 

 of the axis of direction which it meets be considered the positive side of 

 that axis. Let the opposite direction of revolution be called negative. 

 But if any difficulty arise about the use of negative angles, remember 

 that this is merely matter of convenience, and can as well be dispensed 

 with. Four right angles may be added to any angle of direction, with- 

 out altering the direction indicated : and it is perfectly indifferent 

 whether we speak of 160 in the negative direction, or + 200 in the 

 positive direction. 



17. By A = (a, a), let us agree to mean that A signifies a line of a 

 unite of length, inclined at an angle a to the positive side of the axis 

 of length. Use the theoretical mode [ANGLE] of measuring angles. 

 Then (a, a), (a, a 2ir), (a, a i 4ir), &c., are all representations of the 

 same line A. The peculiar symbols of the new algebra, so long as 

 they are wanted, shall be capital letters ; small letters denoting the 

 symbols of the single algebra. 



18. The first four of the fundamental rules in 3 may be easily 

 shown to be true. The geometrical propositions principally required 

 are two. First, that any number of lengths being given and their 

 directions, if we set off from a point through those lengths and in those 

 directions, one after the other, we reach one and the same point in 

 whatever order the lines of progress are taken. Secondly, that if any 

 polygon revolve about one of its angular points, while at the same 

 time the sides and diagonals which meet in that point are all changed 

 in the same ratio, the figure always remains similar to what it was at 

 the outset. With these propositions and the following, 



Co, ol x (b, 8) = (ab, a + /J) $%= (j , -/), 



which are merely expressions of the definitions, there will not be much 

 difficulty in establishing the first four of the rules. 



1 9. Let the square, cube, fourth power, c., of A, as yet, be denoted 

 in full by AA, AAA, AAAA, &c., and their corresponding roots by 

 v' A, y A, J/ A, &c. There is then no difficulty in establishing, as in 

 NEGATIVE, Ac., that the mth power of the nth root of (a, o) is 

 derived from 



I- Ijting any integer ; that ^/ A has n values and no more ; that */ 1 

 stands for a unit on one side (say on the positive side) of the axis of 

 direction, and / 1 for a unit on the negative side. Also, as in 

 the article cited, that, if p and q be the projections of A, on the axis of 

 length and direction, that A is identical in meaning with p q %/ 1 

 and with a (cos o + sin a . / 1). Indeed, at this point it will be 

 advisable for the student to review the imperfect system in NEGATIVE, 

 only reading exponents as is done just above, powers at full length, 

 roots with the old radical sign. 



20. Why do we not hitherto admit the exponent, and define A 2 to 

 mean AA, etc. ? Because we are not prepared with a definition which 

 will include all exponents. In the article just cited, we get the 



meaning of < by interpretation, and it seems only a sort of 



accident that it should have an intelligible meaning. In order to provide 

 beforehand a complete definition of the exponent, which shall make 



(a + b V 1) as fully explained as any other symbol, 



nb iaitio, we must premise knowledge of the arithmetical theory of 

 Naperian LOGARITHMS, and must lay down the definition of a line 

 which answers to, and performs all the functions of, a logarithm. But 

 as it is inconvenient to retain this name, let us substitute for it the 

 word logometer, reserving the word logarithm for arithmetical use. By 

 AA, the logometer of A or (a, o), is meant a line of which the projections 

 on the axes* of length and direction are log a and a ; a line, in fact, of 

 the length V^log a -t- o 2 ) inclined to the unit-line at an angle whose 

 tangent is a-=-log a. Conversely, the line whose logometer is B or 

 (6, ft) has 6 cos ft for the logarithm of its length, and b sin $ for the 

 angle (number of theoretical units in the angle). Every line lias an 

 infinite number of logometers ; for the angle a may be read as oi2niT, 

 where m is any integer ; and thus we have an infinite number of logo- 

 meters, hypothennses to a set of right-angled triangles, whose common 

 basc^s log a on the axis of length. But no logometer belongs to more 

 than one primitive line. 



21. The logometer has the fundamental properties of a logarithm : 

 tlju-: AA + AB=A(AB), meaning that any logometer of A added (bear in 

 mind the extended meaning of all terms of operation) to any logometer 



* Any two perpendicular lines would do ; but to choose any other except the 

 axis of length and direction would be a step precisely equivalent to preferring 

 orae other base to the Naperian in the logarithms of common aJghra. 



of B gives one of the logometers of AB. We shall endeavour to give 

 short heads of demonstration to this and the following propositions. 

 It is readily shown that the projections of two Hues on either axis 

 must be added (with their proper signs) to give the projection of the 

 sum. Now in AA + AB, the sums of the projections are log a-t-log 6 

 and o + $ ; but the projections of A(AB) or \(ab, a. + ) are by definition 

 log (ab) and a + ft whence the proposition follows. 



22. Denoting by e, as usual, the base of Napier's logarithms, it is 

 easy to see that (e, 0) has the simple unit line for one of its logometers, 

 or (1, 0), or 1 ; also that B V 1 is one of the logometers of a unit of 

 length inclined at the angle S, or of (1,9), or of cos 6+ sin 6 . V 1. 

 And by e we must agree to mean (e, o), never (e, 2ir), &c. 



B 



23. Now let A be defined to mean the line which has for one of its 

 logometers B multiplied into any one of the logometers of A : accord- 



B 



ingly A has as many meanings as we can derive different lines by this 

 process from the different logometers of A. And we shall show, first, 

 that the fundamental rules are satisfied by this definition ; second! v, 

 that whenever B means a length of 6 units in the axis of lenyt/i, the 



Li b 



symbol A is exactly the A of common algebra. It is sufficiently 



obvious that A has meaning or meanings for all possible assigned lengths 

 and directions of A and B. 

 A 



24. Since e means the line whose logometer is AA or A, it follows 

 XA BAA B 



that f is the same as A. Hence is the same as A , the line 

 whose logometer is BAA. 



B C\ 



25. The sum of BAA and CAA (the logometers of A and A / 



is (B + C)AA, which is, by definition, the logometer of A . Hence 



B + C B 



A and A A , having the same logometers, are identical. Again, 



A C has for its logometer BAA +BAC or BA(AC), which is also the 



B B 11 B 



logometer of (AC) : hence A C and (AC) are identical. Thirdly, the 



B\C B\ 



logometer ol (A. I is C x A (A ) or C x B x AA, which is also the 



BC / B\C BO 



logometer of A : hence \A ) and A are identical. Consequently 



the fifth fundamental rule is true of A as here defined. 



26. Let B be a line on the axis of length, represented by (4, 0) or 

 (b, IT) according as it is positive or negative. Then BAA is mode simply 

 by multiplying the length of AA by b, and leaving it otherwise un- 

 altered in the first case, or turning it through two right angles in the 

 second. In the first case the projections of BAA are b times those of 

 AA, in the second they have also their signs changed. In the first case 



B B 



then 6 log a and ba. are the projections of the logometer of A , or A is 



\a , ba) , agreeing with A in common algebra : in the second case 



B / It 

 A. is 



/ It \ 



\a , ba), agreeing with A 



27. The meaning of t is the line whose logometer is 8V 1 



x log(*, 0), or fly 1. This line is (1,9), or cos + shift. V 1, 

 whence the equation 



8V i 



e = cos B + sin 9 . V 1 .... (g) 



is a necessary consequence of the various trains of definition. Now, as 

 all we know of these trains of definition is that the meanings of the 

 symbols satisfy the five rules in 3, it may seem to be too much that 

 so remarkable an equation as the last should be actually involved in the 

 definitions, instead of being the result of a long sequence of reasonings. 

 And in truth it is too much in one point ; for since all our preceding 



B 



reasoning on the subject of A would apply equally to any base we 

 might choose for logarithms, and any unit for measuring angles, what 

 have we done but prove the preceding equation true for any base and 



B 



any angular unit ? And we reply that so far as the definition of A is 

 concerned, our proof is perfectly general : but that, on looking back, 

 we find a restrictive connection between the logarithmic base and the 

 angular unit, as follows : It is very easy to see that in our prior 

 definitions, the equation 



{ (1, 9) . (1, ) .(1,9) ... (TO times)} =(1, ,) 

 leads to the following 



Bt 



|cos 9 + sin 6 . / 1| =cos jftfl+sinrnfl . V 1 



in which we may use m as an exponent, since for the simple integer, 

 representing a line in the axis of length, the definition in 23 gives 

 A* =AAA . . . (m times). Let be the angular unit ; then we have 

 (cosl + sinl. / 1) =cos m-t-sin m . V 1. 



mV 1 



But the last is c , if we introduce the complete exponent from 



23 ; therefore it must be an equation of connection between e and the 

 mode of assigning angle 1, that 



TH t/ 1 m 



( should be (cos 1 + sin 1 . V 1) 



