The first yt wrote upon this Subject, in Europe, termed it ye Rule of Restitution & Opposition; Since, it has been called by some, the Analytick Art; by others, Specious Computation; Regula Rei et Census; ye great Art; Modern Geometry; Universal Mathematicks &c. (Diman, p. 1; Langdon, p. 1.) This introduction shows an interest on the part of the author of the original in the history of the subject. For this history he drew directly from John Wallis, or from some work founded on Wallis. In Antiemetical Questions wich numbers as are to be adde abstracted, or any how altered in your forms must be expressed by neces; but such as Pemain unchanged are frequently mixstit recies, & are termed absolute numbers. That & leveral operation in Algebra may be & more clearly ca reft, & following Signs (forst introduced by Neta) are used. Algebraical Characters. Symbols given in the Langdon (1739) manuscript from Harvard College The paragraph cited shows a strong resemblance to the opening paragraph in the article on algebra in Chambers's Cyclopaedia.10 Symbols.-A page of "Algebraical Characters" follows the introduction. Attention is called to the interesting features of this table of symbols; to the bar which is the only sign of aggregation 9 John Wallis. A Treatise of Algebra both Historical and Practical. 10 E. Chambers, Cyclopaedia. Second edition, London, 1738. London, 1685. used; to the symbol for continued proportion; to those for inequality, unequal parallel lines met at one set of extremities by a vertical transversal;11 to the capital S turned on its side to indicate the difference between a and b; 12 a2a5 is explained to mean "a invold to ye 2a x to ye 5th power & joined in one product," although the more common form aa is found, and frequently even such forms as aaaaa, showing the difficulty in adopting the representation by the exponent. Later in the manuscripts, it is curious to find the powers of xy up to the sixth in this latter form, so that the sixth power reads xxxxxx 6 xxxxxy-15xxxxyy—20xxxyyy+15xxyyyy—6xÿÿÿÿÿ+ÿÿÿÿÿ (sic) and then to find the seventh, eighth, and ninth powers in the present day form. (Diman, p. 27; Langdon, p. 19.) Topics. The topics reproduced as they appear in the two manuscripts are set down to show the similarity between them. They are paired except when the topics are identical, a blank space indicating that the topic is omitted. The first member of each pair is taken from the Diman manuscript, the second from that by Langdon. 13 Notation; Algebraical Caracters, Algebraical Characters; Addition of Integers; Subtraction, Substraction; Multiplication of Algebraic Integer, Multiplication of Algebraick Integers; Division; Algebraical Fractions, Algebraick Fractions; Addition & Subtraction of Fractions, Addition & Substraction of Fractions; Multiplication of Fractions; Division of Fractions; Involution of whole Quantities; Involution of Fractional Quantities, Fractional Involution; Evolution of whole Quantities; Fractional Evolution; Binomial Quantities; Involution [of binomial quantities]; Promiscuous Examples [the examples are found but not the heading]; [Heading not given, but two fractions are included in the set of examples], Involution of Binomial Fractional Quantities; Multinomial Quantities; Involution [of multinomial compound quantities]; [No heading but the statement: "Fractional Compound Quantities are also Involved in ye same maner"], Fractional Compound Quantities; Evolution, Evolution of Multinomial Quantities; Surd Quantities; Notation [of surds]; Reduction of Surds; Multiplication of Surds; Division of Surds; Addition and Subtraction of Surds, Addition and Substraction of Surds; Compound Surds; Multiplication of Binomial Surds, ; Division in Compound Surds, -; Equation; Reduction of Equations; Reduction by Addition; Reduction by Subtraction, Reduction by Substraction; Reduction by Multiplication; Reduction by Division; Reduction by Involution; Reduction by Evolution; Reduction by Analogies to Equations & e Contra; The Method of Resolving Algebraical Questions; General Rules Concerning ye Reduction of Equations; Simple Equations; The Solution of Adfected Quadratick Equations, ; Mr Oughtreds method of solving adfected Quadraticks, Mr. Oughtreds method of solving adfected Quadratics; The Solution of Adfected Equations by taking away ye Second Term, The Solution of adfected Equa 11 These symbols are used by William Oughtred in his Clavis Mathematica, p. 166, London, 1648. 12 This symbol was used first by Oughtred, loc. cit., Eu. 2, 1652. 13 Diman uses the spelling "subtraction," while Langdon uses "substraction." The latter is used throughout Greenwood's Arithmetic. For a discussion of the spelling of these words, see David Eugene Smith, History of Mathematics, 2 vols., Boston, 1923-24, hereafter referred to as Smith, History, II, 95. tions by taking away the second term; The Solution of Adfected Quadratick Equations by ye method of Compleating ye Square, The Solution of Adfected Quadratic Equations by y Method of compleating ye square; Questions, Questions producing adfected Quadratick Equations; The Resolution of Cubic Equations, The Resolution of Cubick Equations; Cubic Equations by Substitution, Cubick Equations by Substitution; Cubic Equations by Tryalls & Depression, Cubick Equations by Tryals & depression; The Solution of Irregular Cubics, The Solution of irregular Cubicks; The Method of Converging Series, The method of converging Series and Approximation; —, I of Simple Roots; II of Adfected Equations; Mr Raphson's Theorems for Simple Powers [not so designated, but all the Raphson formulas are given]; Mr. Raphson's Theorems for adfected Equations; Dr Halley's Theorems for Solving Equations of all sorts, Dr. Halley's method for solving equations of all sorts; Concerning the Method of resolving Geometrical Problems algebraically, Concerning yo method of solving Geometrical Questions Algebraically. Treatment of topics.-Many interesting passages show the spirit and subject matter of these two notebooks, and at the same time multiply the evidence bearing on their common source. Some important ones will be touched upon. The clearness of explanation throughout may be illustrated by the treatment of signs in addition and subtraction, which begins as follows: The reason of y° Operation in Algebraical Addition and Subtraction may be easily understood by considering ye affirmative and negative Quantities like opposites as y Case is in Ballancing accompts. (Diman, p. 7; Langdon, p. 5). Involution of Binomial Quantities. . . . Consequently if ye Numeral figures of Coefficients could be found ye whole might be performed without multiplication and this is done by ye following Problem. To find y Coefficients in Binomial Powers. Rule. Multiply y Coefficient into ye Index of y® Power and Divide that Product by ye Number of terms, counting from y° left hand, and ye Quotient will be ye Coefficient or Numeral Figure of yo next successive Quantity. (Diman, p. 27; Langdon, p. 18). Irrational Quantities are noted thus: √2 weh is 2 wth ye Sign of Irrationality before it. . . . There is also another way of marking surd Quantities where Roots are expressed without ye Radical sign by their Index, this is founded upon ye manner of expressing Powers, thus as x2, x3, x* signifies y Square, Cube & Biquadratick of : so x, x, x will accordingly signify y Square, Cube and Biquadratick Root of x. . . . and w" at any time there is y Sign of Irrationality prefixt to mixt Quantities with ye sign of Inseperation (sic) over ym thus: √37+V:2 it is called a universal Root. (Diman, p. 33f; Langdon, p. 23f.) There are seven methods of "Reduction of Equations." The seventh, "Reduction of Analogies to Equations & e Contra," is illustrated by Ex. 1: Reduce ye Analogy x 4: 2x, 4 × 416, x × 2x = 2xx, Euclid, xx=8 (Diman, p. 59; Langdon, p. 31). 2xx 16 =8 per 16 : 6 2 2 Under "The Method of Resolving Algebraical Questions" This part of Algebra is wholly arbitrary & everyone is left to himself to pursue his own particular Genius and way of thinking, which is so far from being a Defect yt it is one of ye Chief Excellencies of this Science, which may from hence not unjustly be called a sublime way of Reasoning. (Diman, p. 62; Langdon, p. 33). Eight rules precede the set of questions under " Simple Equations," and these are accompanied by illustrative examples, employing letters throughout. The method employed in the questions can best be understood by examining one of them. This one bears the same number in the two manuscripts. The Refolution of Cubic Equations. 1. A Cubic Equation has for & first Term & Cube of you unharown Quantity, for y Second Quantity multiphed into a foeffunt, dll. Ex: xex+red +xg = 3714544. Junlrow Quantity =x d=120.300 = g... 2. A Cubic Equation is formed from one two or threekt. as nomials involed together! representing a Compleat Cube or Parallellepiped whose Base is wither a Goometrical Square or Parallellogram: 3. To find of Value ox. Algebraick Computation. 1. By Substitution, Deduction, & Division_. 2. By trial and then Depressing it to a Square! Introduction to cubic equations in the Diman (1730) manuscript from Harvard College Quest. 25. I am a Brazon Lyon, my two Eyes, my Mouth & ye Sole of my Right foot, are so many several Pipes, which fill a Cistern, yo Right Eye in 2 Days, y Left in 3, & y Sole of my foot in 4, but my Mouth can fill it in 6 hours, tell me in wt. time all these to-gether, my mouth, my Eyes and my foot will fill y Cistern. (Diman, p. 77; Langdon, p. 41.) х х х х The unknown x is taken for the number of hours sought, and for the part of the cistern filled by the respective pipes. a b c d From these fractional values an equation is formed and a general -" -" solution in terms of a, b, c, d effected. In the value of a so found, numerical substitutions are made. This is the usual procedure in the solution of the questions. Diman gives 26 such problems to be solved, while Langdon gives 30. Twenty-two problems are alike in the two works. The problems are mixed in their nature, Diman favoring somewhat the more mechanical kind and Langdon the more practical-practical for the day in which they arose, if not for theirs nor for ours. They include age, merchants trading for linen and pepper, numbers multiplied, divided, and operated upon in a variety of ways, the vintner, the man who found poor persons at his door ready to receive alms, clocks, the shepherd in time of war, cisterns, noblemen traveling for pleasure, the gentleman who hired a servant, a general setting his army in square array, two persons discoursing about their money, and partnership. "Adfected Quadratick Equations" are considered under three forms, and "Each of these Forms may be Resolv'd 3 several ways." The first of the methods shown is: Mr. Oughtreds 14 Method of Solving Quadraticks. . . Rule. Multiply y° absolute Number by four & add thereunto ye Square of y° Differential Quantity, ye Square Root of ye minus ye Differential Quantity being divided by two, is ye Quantity Sought. (Diman, p. 78; Langdon, p. 43). The second method is "The Solution of Adfected Equations by taking away ye Second Term." (Diman, p. 81; Langdon, p. 46.) In this method y-12 d is substituted in the equation x2+dx=m; whence y2-yd+1⁄2 d2x2 and y2=m+1⁄4/ d2 or y=√m+1⁄4 d2. The labor involved in removing one term is admissible only from the standpoint of the interest inherent in a different manner of solution. The third of the three methods given for the treatment of the quadratic equation is "The Solution of Adfected Quadratick Equations by ye method of compleating ye Square." This method is the familiar one known by the same title at the present time. A set of 25 problems follows these three methods, and the two manuscripts agree in the problems and in the order of them with a single exception. It is of especial interest to note that, in connection with the solution of question 7 of this set, the Langdon manuscript (p. 49) gives an imaginary number in the result. This is the only approach to an imaginary in either of the manuscripts, and indicates scholarship on the part of the author of the notes as well as ability on the part of his pupil. 14 Oughtred's Clavis Mathematica consists for the most part of the solution of quadratic equations. On his treatment of these equations, see F. Cajori, William Oughtred, p. 29, Chicago, 1916. 95337-24-2 |