A PROPOSAL About Printing a TREATISE of ALGEBRA, HISTORICAL and PRACTICAL: Written by the Reverend and Learned Dr. John Wallis (Savilian Professor of Geometry in the University of Oxford), containing not only a History, but an Institution of ALGEBRA, according to several Methods hitherto in practice; with many Additions of his own.  
IT contains an Account of the Original, Progress, and Advancement of (what we now call) Algebra, from time to time, and by what steps it hath attained to that height at which now it is.  
Asserting, That it was in use of old among the Grecians; but studiously concealed as a great Secret.  
Examples we have of it in Euclid, at lest in Theon, upon him; who ascribes the invention of it (amongst them) to Plato.  
Other Examples we have of it in Pappus, and the effects of it in Archimedes, Apollonius, and others, though obscurely covered and disguised.  
But we have no professed Treatise of it (among them) ancienter than that of Diophantus, first published (in Latin) by Xylander, and since (in Greek and Latin) by Bachetus, with divers Additions of his own; and reprinted lately with Additions of Monsieur Fermat.  
That it was of ancient use among the Arabs (and perhaps sooner than amongst the Greeks), which they are supposed to have received from the Persians, and these from the Indians.  
From the Arabs (by means of the Saracens and Moors) it was brought into Spain, and thence into England (together with the use of the Numeral Figures, and other Parts of Mathematical Learning, and particularly the Astronomical), before Diophantus seems to have been known amongst us. And from those we have the name of Algebra.  
And indeed most of the Greek Learning came to us the same way, the first Translations of Euclid, Ptolemy, and others, into Latin, being from the Arabic Copies, and not from the Greek Originals.  
That the use of the Numeral Figures (which the Greeks had not) was a great advantage to the improvement of Algebra.  
These Figures seem to have come in use in these Parts about the middle of the Eleventh Century (about the year 1050), though some others think not till about 200 years after, and it seems they did scarce come to be of common use till about that time.  
The Sexagesimal Fractions (introduced it seems by Ptolemy) did but imperfectly supply the want of such a Method of Numeral Figures.  
The use of Numeral Figures have received two great Improvements, The one is that of Decimal Parts, which seems to have been introduced (silently and unobserved) by Regiomontanus, in his Trigonometrical Canons, about the year 1450, but much advanced in the last and present Century, by Simon Stevin and Mr. Briggs, etc.  
And this is much to be preferred before Ptolomy's Sexagesimal way, as is showed by the comparative use of both.  
The other Improvement is that of Logarithms, which is of great use in Astronomical and other Trigonometrical Calculations, introduced by the Lord Neper, and perfected by Mr. Briggs (about the beginning of this Century). The ground and practice of which is here declared.  
And these things, though they be not properly Parts of Algebra, are yet of great advantage in the practice of it.  
The first printed Author which treats of Algebra is Lucas Pacciolus, or Lucas de Burgo a Minorite Friar, of whom we have a Treatise in Italian, printed at Venice in the year 1494, (soon after the first Invention of Printing.)  
But he therein mentions Leonardus Pisanus, and divers others more ancient than himself, from whom he learned it, but whose Works are not now extant.  
This Friar Lucas, in his Summa Arithmetica & Geometrica, (for he hath other Works extant) hath a very full Treatise of Arithmetic in all the Parts of it; in Integers, Fractions, Surds, Binomials; Extraction of Roots, Quadratic, Cubic, etc. and the several Rules of Proportion, Fellowship, about Accounts, Alligation, False Position; and of Algebra, with the Appurtenances thereunto, as far as Quadratic Equations reach, but no farther: And this he tells us was derived from the Arabs, (to whom we are beholding for this kind of Learning,) without taking notice of Diophantus (or other Greek Authors), who it seems was not known here in those days.  
After him followed Stiphelius (a good Author), and others by him cited, who also proceed no farther than Quadratic Equations.  
Afterwards Scipio Ferreus, Cardan, Tartalea, and others, proceeded to the Solution of (some) Cubic Equations.  
And Bombelli goes yet farther, and shows how to reduce a Biquadratic Equation (by the help of a Cubic) to two Quadratics.  
And Nonnius (or Nunnez, in Spanish) Ramus, Schonerus, Salignacus, Clavius, and others, (in the last Century) pursued the same Subject, in different ways; but (for the most part) proceed no farther than Quadratic Equations.  
In the mean time, Diophantus, first by Xylander (in Latin), and afterwards by Bachetus was made public, whose Method differs much from that of the Arabs (whom those others followed), and particularly in the order of denominating the Powers; as taking no notice of Sursolids, but using only the names of Square and Cube, with the Compounds of these.  
And hitherto no other than the unknown Quantities were wont to be denoted in Algebra by particular Notes or Symbols, but the known Quantities by the ordinary Numeral Figures.  
The next great step, for the improvement of Algebra, was that of Specious Arithmetic, first introduced by Vieta about the year 1590.  
The Specious Arithmetic, which gives Notes or Symbols (which he calls Species) to Quantities both known and unknown, furnisheth us with a short and convenient way of Notation; whereby the whole process of many Operations is at once exposed to the Eye in a short Synopsis.  
By help of this he makes many Discoveries, in the process of Algebra, not before taken notice of.  
And he introduceth his Numeral Exegesis, of affected Equations, extracting the Roots of these in Numbers. Which had before been applied to single Equations, such as the extracting the Roots of Squares, Cubes, etc. singly proposed; but had not been applied to Equations affected. And in the Denomination of Powers, he follows the order of Diophantus, not that derived from the Arabs, which others had before used.  
The method of Vieta is followed and much improved by Mr. Oughtred in his Clavis (first published in the year 1631) and other Treatises of his; and it doth in a brief compendious method declare in short what had before been the Subject of large Volumes.  
And for this reason, here is a pretty full account of his Method inserted, together with an Institution for the practice of Algebra according thereunto; and though much of it had been before taught in the Author's abovementioned, yet this was thought the most proper place to insert such an Institution, because by him delivered in the most compendious form.  
And in pursuance of his Method, and as an Exemplification thereof, there is here inserted a Discourse of Angular Sections, and several things thereon depending.  
Mr. Harriot was contemporary with Mr. Oughtred, but died before him, and left many good things behind him in writing, of which there is nothing hitherto made public, but only his Algebra or Analytice, which was published by Mr. Warner soon after that of Mr. Oughtred, in the same year 1631.  
He altars the way of Notation, used by Vieta and Oughtred, for another more convenient.  
And he hath also made a strange improvement of Algebra, by discovering the true construction of Compound Equations, and how they be raised by a Multiplication of Simple Equations, and may therefore be resolved into such.  
By this means he shows the number of Roots (real or imaginary) in every Equation, and the Ingredients of all the Coefficients, in each degree of Affection.  
He shows also how to increase or diminish the Roots, yet unknown by any Excess, or in any Proportion assigned; to destroy some of the intermediate Terms, to turn Negative Roots into Affirmative, or these into those; with many other things very advantageous in the practice of Algebra.  
In sum, He hath taught (in a manner) all that which hath since passed for the Cartesian method of Algebra, there being scarce any thing of (pure) Algebra in Des Cartes, which was not before in Harriot, from whom Des Cartes seems to have taken what he hath, that is purely Algebra, but without naming him.  
But the Application thereof to Geometry, or other particular Subjects, is not the business of that Treatise, (but what he hath handled in other Writings of his, which have not yet the good hap to be made public;) the design of this being purely Algebra, abstract for particular Subjects.  
Of this Treatise here is the fuller account inserted, because the Book itself hath been but little known abroad; that it may hence appear to what estate Harriot had brought Algebra before his death.  
After this follows an account of Dr. Pell's Method, who hath a particular way of Notation, by keeping a Register (in the Margin) of the several steps in his Demonstrations, with References from one to another.  
Of this, some Examples are here inserted of his own, and others in imitation thereof; with intimation how that innumerable Solutions of undetermined Cases are by his method easily discoverable, where great Mathematicians have thought it a great work to find out some one.  
On this occasion there is a farther Discourse of Vndetermined Questions, and the Limitation of them, and particularly of the Rule of Alligation, and of (what they call) Geometrical Places, which are of a like nature, and but the Geometrical Construction of (some of) these Undetermined Questions.  
After this is a Discourse of Negative Squares, and the Roots of them, on which depend (what they call) imaginary Roots of impossible Equations, showing what is the true Import thereof in nature, with divers Geometrical Constructions suiting thereunto.  
Then follows a Discourse of the Method of Exhaustions (used by Ancients and Moderns), with the foundation of it.  
And in pursuance thereof, the Geometrica Indivisibilium of Cavalerius, showing the true import thereof, and its agreement with the Ancients Method of Exhaustions, as being but a compendious Expression thereof, and grounded thereupon.  
Consequent to this, is the Arithmetica Infinitorum, which depends also on the method of Exhaustions; taking that to be equal, which is proved to differ by less than any assignable Quantity.  
And lastly, the method of infinite Series, or continual Approximations, (grounded on the same Principles) arising principally from Divisions and Extraction of Roots, in Species, infinitely continued; invented by Mr. Isaac Newton, and pursued by Mr. Nicholas Mercator, and others, which is of great use for the rectifying of Curve Lines, squaring of Curve-lined Figures, and other abstruse Difficulties in Geometry.  
Several other Discourses are in several places inserted; as, Of Aliquote Parts, and other Questions depending thereon; and divers other particulars, which will be seen in the Work itself.  
The whole being written in English, is submitted to the Royal Society, to be printed or otherwise disposed of as they please; and if printed, will contain (as is supposed) about three or four Quires of Paper.  

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