William George Horner (1819). "A New Method of Solving Numerical Equations of All Orders, by Continuous Approximation". Philosophical Transactions of the Royal Society of London.
"For this- o ; r, r, &c. -= i or . 1 &c. and purpose, we might assume R adopt, as most suitable to these conditions, the algorithm of Theorem II, until we had arrived at R", an unambiguous Jinmitingvalue of x."
"The abbreviations used in the close of this example, are of a description sufficiently obvious and inartificial; but in order to perfect the algorithm of our method in its application to higher equations, and to the progress by simple digits, attenetion must be given to the following general principles of Compendious Operation"
William Rowan Hamilton (1844). "On a new species of imaginary quantities connected with a theory of quaternions". Proceedings of the Royal Irish Academy.
"A new sort of algorithm, or calculus, for spherical trigonometry, appears to be thus given, or indicated."
M Davies (1848) "XLIX. Analysis of the theory of equations, with a few remarks on recent English work on the subject."
"The criterion of Budan in the form in which he himself presented it, is entirely useless ; and I am bold to say that it would not have attracted the least notice but for the form in which Mr. Homer arranged the algorithm and explained the ..."
"Geometry, trigonometry and statics, are laid under contribution to demonstrate relations which will be seen to flow as immediate and obvious consequences from the most elementary principles in the algorithm of determinants."
No access to full text.
Difficult to judge without access to the full text, but here "algorithm of determinants" seems to be used in the sense of "calculus of determinants" instead of "an algorithm to compute determinants".
J. J. Sylvester (1879). "Note on Determinants and Duadic Synthemes". American Journal of Mathematics.
"In the subjoined table I have calculated the values of the residues of these numbers by a regular algorithm in respect to moduli beginning with 223 and regularly decreasing according to the descending powers of 2."
Robert Potts (1879). Elementary algebra, with brief notices of its history
Algorism
Henry Burchard Fine (1891). The Number-system of Algebra: Treated Theoretically and Historically
"Europe in turn derived the system from the Arabians in the 12th century, the " Gobar " numerals (table, column 5) of the Arabians of Spain being the pattern forms of the European numerals (table, column 7). The arithmetic founded on the new system was at first called algorithm (after Alkhwarizmi), to distinguish it from the arithmetic of the abacus which it came to replace."
Used in the meaning of "algorism", not "algorithm".
Eric Doolittle (1912). "The Secular Variations of the Elements of the Orbits of the Four Inner Planets Computed for the Epoch 1850.0 G. M. T.". Transactions of the American Philosophical Society
"The work of GAUSS contains no application to the determination of secular variations nor are all the formulas necessary for this purpose there developed; the first integration alone is effected, and it is shown that by changing first to the variable E' and afterward introducing a new variable, T, each of the complicated integrals may be made to depend upon elliptic integrals whose values GAUSS obtained by the introduction of a new algorithm called by him the Arithmetico-geometrical mean."
"... rules of numeral computation, that is, addition ..."
1743: Edmund Stone: A New Mathematical Dictionary (Second Edition)
1795: Charles Hutton, A Philosophical and Mathematical Dictionary
"ABACUS, in Arithmetic, an ancient instrument used by most nations for casting up accounts, or performing arithmetical calculations. It is by some derived from the Greek a?a£, which signifies a cupboard or beaufet, perhaps from the similarity of the form of this instrument: and by others it is derived from the Phoenician abak, which signifies dust or powder, because it was said that this instrument was sometimes a square board or tablet, which was powdered over with fine sand or dust, in which were traced the figures or characters used in making calculations, which could thence be easily defaced, and the abacus refitted for use. But Lucas Paciolus, in the first part of his second distinction, thinks it is a corruption of Arabicus, by which he meant their Algorithm, or the method of numeral computation received from them."
"ALGORITHM, is similar to logistics, signifying the art of computing in any particular way, or about some particular subject ; or the common rules of computing in any art. As the algorithm of numbers, of algebra, of integers, of fractions, of surds, &c ; meaning the common rules for performing the operations of arithmetic, or algebra, or fractions, &c."
1814: Peter Barlow, A New Mathematical and Philosophical Dictionary
