The equations to the asymptotes are = t y/b and x = =y respectively.
Among the great variety of problems solved are problems leading to determinate equations of the first degree in one, two, three or four variables, to determinate quadratic equations, and to indeterminate equations of the first degree in one or more variables, which are, however, transformed into determinate equations by arbitrarily assuming a value for one of the required numbers, Diophantus being always satisfied with a rational, even if fractional, result and not requiring a solution in integers.
But the bulk of the work consists of problems leading to indeterminate equations of the second degree, and these universally take the form that one or two (and never more) linear or quadratic functions of one variable x are to be made rational square numbers by finding a suitable value for x.
A few problems lead to indeterminate equations of the third and fourth degrees, an easy indeterminate equation of the sixth degree being also found.
Often assumptions are made which lead to equations in x which cannot be solved "rationally," i.e.
Laplace (1801); Traite analytique des courbes et des surfaces du second degre (1802); Recherches sur l'integration des equations differentielles partielles et sur les vibrations des surfaces (1803); Traite de physique (1816); Recueil d'observations geodesiques, astronomiques et physiques executees en Espagne et Ecosse, with Arago (1821); Memoire sur la vraie constitution de l'atmosphere terrestre (1841); Traite elementaire d'astronomie physique (1805); Recherches sur plusieurs points de l'astronomie egyptienne (1823); Recherches sur l'ancienne astronomic chinoise (1840); Etudes sur l'astronomie indienne et sur l'astronomie chinoise (1862); Essai sur l'histoire generale des sciences pendant la Revolution (1803); Discours sur Montaigne (1812); Lettres sur l'approvisionnement de Paris et sur le commerce des grains (1835); Mélanges scientifiques et litteraires (1858).
Since the distance of a body from the observer cannot be observed directly, but only the right ascension and declination, calling these a and 6 we conceive ideal equations of the form a = f (a, b, c, e, f, g, t) and 5=0 (a, b, c, e, f, g, t), the symbols a, b,.
If the values ofa and 6, defining the position of the body on the celestial sphere, are observed at three different times, we may conceive six equations like the above, one for each of the three observed values of a and S.
Then by solving these equations, regarding the six elements as unknown quantities, the values of the latter may be computed.
For Tartaglia's discovery of the solution of cubic equations, and his contests with Antonio Marie Floridas, see Algebra (History).
His largest work,Trattato generale di numeri e misure, is a comprehensive mathematical treatise, including arithmetic, geometry, mensuration, and algebra as far as quadratic equations (Venice, 1556, 1560).
This section treats of such subjects as nomenclature, formulae, chemical equations, chemical change and similar subjects.
We cannot deal with equations that big—but a computer will solve for that in a minute if it has enough data.
Why, I can do long, complicated quadratic equations in my head quite easily, and it is great fun!
Only then, expressing known historic facts by equations and comparing the relative significance of this factor, can we hope to define the unknown.
And by bringing variously selected historic units (battles, campaigns, periods of war) into such equations, a series of numbers could be obtained in which certain laws should exist and might be discovered.