For an **algebraic** solution the invariants must fulfil certain conditions.

It was then proposed to arrange a detector so that it was affected by the **algebraic** sum of the two oscillations, and by swivelling round the double receiving antennae to locate the direction of the sending station by finding out when the detector gave the best signal.

Under the general heading "Algebra and Theory of Numbers" occur the subheadings "Elements of Algebra," with the topics rational polynomials, permutations, &c., partitions, probabilities; "Linear Substitutions," with the topics determinants, &c., linear substitutions, general theory of quantics; "Theory of **Algebraic** Equations," with the topics existence of roots, separation of and approximation to, theory of Galois, &c. "Theory of Numbers," with the topics congruences, quadratic residues, prime numbers, particular irrational and transcendental numbers.

Evolution and involution are usually regarded as operations of ordinary algebra; this leads to a notation for powers and roots, and a theory of irrational **algebraic** quantities analogous to that of irrational numbers.

The name l'arte magiore, the greater art, is designed to distinguish it from l'arte minore, the lesser art, a term which he applied to the modern arithmetic. His second variant, la regula de la cosa, the rule of the thing or unknown quantity, appears to have been in common use in Italy, and the word cosa was preserved for several centuries in the forms toss or algebra, cossic or **algebraic**, cossist or algebraist, &c. Other Italian writers termed it the Regula rei et census, the rule of the thing and the product, or the root and the square.