Of late years another theory or impossible quantities has been brought forward by NI. Bue:, in the London l'hil. Trans. 1805, also by M. Argaud, in a work with this title, .Essai sur one inani-re de re-presenter quantites imaginaires dans les constructions gcumetriques, published in France in 1806, and again by M. J. F. Francais, in Nouveaux prin. cipes de geometric de position et interpretation geome trique des symbols published in Annales de ilaihematiques. Sept. 1813. According to these writers, the impossible character ,/-1, is the sign of perpendicu larity ; so that the imaginary expression a instead of being the sign of an operation which cannot be perform ed, and which has no geometrical representative, is, ac cording to these writers, represented geometrically by a perpen.ficular to a straight line : that is, if a line to the right be expressed by a, and an equal line to the left by a, then a third line perpendicular to and equal to either of these will be expressed by a V-1.
To prove this, the last mentioned writer defines the ratio of magnitude to be the numerical ratio between the magnauchs or the two lines and the ratio of position, the inclination of the one line to the other, or the angle they contain. Again, he lays it down, that four straight lines are in proportion of magnitude and position, when between the two last there is the same ratio of magnitude and po sition as between the two first. That is, supposing a, b, c, d, to be the magnitudes, we must have b -- also d angle contained by a and b equal to the angle contained by c and d.
When the consequent of the first ratio is the antecedent of the second, the proportion of magnitude and position is said to be continued, and the middle term is a mean pro portional of magnitude and position between the other two. From this it follows, that the middle term bisects the an gle made by the two extremes.
These observations being premised, he gives the follow ing theorem as the foundation of his theory. Imaginary quantities of tl.e form a 4/-1 represent in the geome try of position perfiendirulars to the axis of the abscissa, and reciprocally. perpendiculars to the axis of the abscissa are imaginaries of this form. For, putting + a and a for straight lines lying in opposite directions, according to M. Francais, the quantity = a I is a mean proportion al of magnitude and position between them. Now it has been premised, that a line, which is a mean proportional in magnitude and position between two lines, ought to bi sect the angle they contain ; therefore, in the present case, the mean must be perpendicular to the axis of the ab cissx, and will lie above and below the axis according as it is + a ,/-1, or Reciprocally, every per pendicular to the axis of the abcissae must, according to the same principle, be a mean proportional between a and ; it is therefore an imaginary quantity of the form Such is the substance of M. Francais' demonstration ;
but to us it seems to be by no means satisfactory. \Ve should have supposed, that in seeking the mean of magni tude and position between a and a, he would have sought the mean of magnitude independently of position, and then the mean of position independently of magnitude. These would have required different operations ; the first would have given ,/ a x a =a for the mean of magnitude, and, putting q for a right angle, the second would have giv en = q for the mean of position. We cannot, however, see any useful conclusion deducible from the re sult. The author of the theory, by calling the lines -1- a and a, seems to have invested them at once with mag nitude and position, while at the same time he seeks the mean by a process, which applies to them as things hav ing only magnitude.
The proof offered by Mr. Bue in support of the truth of the proposition, that is the sign of perpendiculari ty, is not inure conclusive. lie supposes three equal straight lines to meet in a point, two of them to be in one straight line, and the third to be at right angles to them both. lie calls the line taken to the right + 1, then that taken to the left, he says, must Le l ; and the third, which must be a mean proportional between them, must be ,/(1'), or more simply 4/-1. Hence he infers, that ,jI is the sign of perpendicularity. The inconclusive ness of this reasoning, has been well exposed by an able critic in the Edinburgh Review, vol. xii. July 1808, where it is observed, that any imaginable conclusion might have been derived in the same manner. For example, the third line, instead of being at light angles, may be supposed to make an angle of 120° with the one, and of 60° with the other, and still it would be a MCA!) proportional between the other two ; and its squat c would have been 4- I x = I, and the line itself V 1. It is farther ob served, that it would be very unfortunate for science, and productive of inextricable confusion in mathematical lan guage, if the character which denoted impossibility at one time, should at another signify something actually existing like perpendicularity, which, besides, is not a quantity, but the modification of quantity. It would therefore be strange indeed, if a character which applies only to quantity, whe ther as possible or not possible, should pass to the expres sion of something, of which quantity cannot be predicated.