ons; and I doubt not this fimple Table or Index to the Cafes of the Wafte-Books, what is done. I have alfo in the II. Chapter of the Appendix added feveral useful Inftructions You'll find, it is true, feveral Things faid in the Inftructions, of which I have given no Examples, but I did not fee the Neceffity of examplifying every Thing: By the Examples that are made, the reft will be very easily applied; for I have endea- voured to manage the Explication of the general Rules, fo that by a due Confide- ration of the fe, and the Applications made, you'll be able to apply the Method to any other Cafe or Subject as far as the Nature of the Thing will admit. Oblerve 1. There is a Question omitted to be fet down and wrought in Page 56, and yet in Page 58, it is referred to as Queftion 3d, and stated thus, 7 Ounce .161.. 9 Ounce, the Question is, If I get 7 Ounce Bread for a Penny, when Wheat is at 16 7. the Boll, what is the Price of the Wheat when I get 2. At Page 76, Line 35, you have a Rule relating to fome Circumftance of the Therefore in fuch Cafes, &c. you must alter preceeding Question, it begins,- out fo, fubftra&t the Payment from the Interest due at the Time, and keep the Remainder to be added to the next Sum of Intereft which is calculate upon the preceeding principal Sum, to the Time of the next partial Payment; and if the Payment is greater than the forefaid Sum of Intereft, add the Principal, and that Intereft, and then fubftract; otherwife fubftract from the Sum of Intereft alone, and fo for every fucceeding Part of the Work. 3. In the Leger No. I. you'll find in fome Places, that the Spaces of the Creditor and Debtor Side of the Accompt, do not exactly answer, particularly in Folio 6th, It is neceffary you correct the following ERRATA with your Pen before you begin to read. AGE Line for this very read thus every. 1. 5. put the Parenthefis before each. p. 12. PA.. from by another Line 1. from another, by a Line. p. 25.1. 16. for 10 1.00. 8. 2011. conne&t thus with Number, and begin the next Sentence with if the. p. 34, 1, 38. Conftitution 1. Conftruction. p. 37. 1. 23. any other r. another. p. 44. 1. 1. for 19 r. 17. p. 48. l. 37, Proportion 5th r. Propofition 4th. p.49. 1. 25. add 5 and one 1. 5 add 1. p.63. 1. 17. close the Parenthesis after Queftions. p. 86. 1. 40. for 110 l. r. 100l. p. 99. l. 10, r. . p. 111. l. 11 and 13. for 4.1.14. for 200 r. 20. p. 115. fit r. fit. p. 118. 1. 15. appiled r. applied. p. 122. 1. 29. Congnation r. Confignation. p. 124. 1. 4. after Right dele of. 1. 23. Eaftte 1. Eftate. p. 1371. 43. Rechange 1. Recharge. p. 169. l. 17. the advance: 1. they advance, p. 174. 1. 33. woth 1. doth; and 1. 35. dele to. Appendir, p. 2, 1. 10. for 6 r. 3, forr., 1. 11, for 8 r. 16. p. 7. l. 18. aleady r. already. 1. 43. for 21:6247696 r. 1.26247696. p. 12. 1. 38. Proportion 3d r. Propofition 3d. p. 23. 1. 3. for 1. 1. p. 26. 1. 28. for C: S:: 4: S read C: S :: A: S There are other fmall Efcapes, particularly in the Pointing; but I'm confident there are none fo great as will hinder you to know the true Senfe, and therefore I expect an easy Excuse. Containing a Definition of Arithmetick, its OBJECT and OPERATIONS in general; and particularly NOTATION or NUMERATION. I. W I. DEFINITIONS. は HEN we confider any Thing as One, or alone of its Kind; and then confidering it as indivifible, or at least undivided; or laftly, confidering feveral things as if they were really united, neglecting the Difference that may be among them; the Idea we have of this Thing or Collection of Things, confidered after fuch a Manner, is called UNITY, i. e. one individual Thing of a particular Kind and Name; as one Man, one Stone, one Army, one Kingdom, &c. 2. WHEN WE confider more things together, (which feparately taken we would call Unities or individual Things, according to the firft Definition) whether they are of the fame or different Kinds; looking at no more in them but that they are not one and the fame Thing or Unit; the Idea we have of them confidered in this Manner is called NUMBER; thus we have a Number of Men, a Number of Trees, a Number of Towns, or Armies, &c. in which Examples, the Individuals are all of one Name: But if we fhould conceive together one Horfe, one Man, and one Tree, they make as truly a Number of Things as the other, tho they are different Species of Things. A 3. The 3. THE Science that contemplates the Nature and Properties of Numbers is called ARITHMETICK; the practical Part whereof is the applying these Properties of Numbers to things that occur in human Affairs. From thefe Definitions clearly follow thefe Things, viz. 1. THAT Number is nothing elfe but a Collection or Multitude of Unities: Tho' Unity it felf may improperly be called Number as far as it may answer the Queftion, How many? In which Senfe even Nothing is a Number. 2. NUMBERS are indefinite; for they can be encreafed as long as we can fuppose another Unit added to the former Collection or Number. 3. THE moft general Property of Number confifts in being capable of Encreafe or Decrease by the continual adding or taking away Unity, until the Sum is infinite (if poffible) or Unity remains; which if you alfo take away nothing remains. All other particular Properties, are only the Effect of comparing dif ferent Numbers. I won't here engage very far in the Difpute, whether Unity be properly a Number, otherwife than as I have faid above: If any Body is not pleafed to diftinguish Unity and Number, as I have done, I don't quarrel, let them confider Unity as the firft and leaft Number: Only know, that I have always conceived Number, as implying a Multitude; and Unity as the moft fimple constituent Part thereof: And whenever Unity is called a Number, I take it to be fo, only as it is an Answer to the general Queftion, how many? (i, e. what Number?) yet it is but improperly a Number; and if there are none or not more than one, I think the proper Anfwer is, there is no Number, there is but one, or none: But it were needlefs to give us this Trouble in ordinary Difcourfe: Before the Queftion is asked, it is fuppofed we know not what there is, whether one, or none, or many; and fo as the Question is general, having an equal Refpect to fome or none; the Answer is always fimply, there is one, none, or fo many: As if the Queftion had been, is there one, or more, or none? Inftead whereof we fay more conveniently how many? Comprehending all these Senfes. Now, if the being an Answer to that Question, does not properly make none, a Number, it can as improperly make Unity fo. A certain Author argues thus, viz. If Unity is not a Number, then from any Number take away Unity, the Jame Number remains, which is abfurd. But this is precarious, and only follows from his own Definition of Unity, which he will have to be, a particular Number: It is plain, if according to him a great Number is compofed of leffer Numbers, that if no Number is taken away, the fame Number remains; but if we conceive every Number to be compofed of Unities, then there is no fuch Confequence; for as the adding, fo the taking away Unity, ftill changes the Number. To have done, take it any Way you will, fo you have a clear Notion of the Object of Arithmetick: And for Conveniency I fhall in all that follows confider Unity, as a particular, (tho' improper) Number. 4. IT follows, that all the Operations in Arithmetick are reduceable to two, viz. Augmenting, and Diminishing. 1. § II. Notation, or the Expreffion of Numbers. No UMBERS being indefinite, neither could the Theory of them be examined, nor that be made of any Ufe to Mankind, until a Way was found to exprefs them by certain Signs, whereby one Number might be represented diftinctly from another: Thefe Signs are indeed all arbitrary, but neceffary to be known and agreed upon by Perfons who would understand one another: They are of two Kinds. 1ft, Are certain articulate Words, as One, Two, Three, &c. different in different Languages. 2d, Are certain Marks or Characters used in writing, and agreed upon at this Day by the greater Part of the known World, as 1, 2, 3, 4, &c. Both thefe Ways ought to be known; for as both of them reprefent the fame Ideas in our Mind, fo they are mutual Reprefentatives of one another; thus 3 fignifies Three, and contrarily: But the last being the far fhortest and most convenient Expreffion, in order to all the Operations that are to be performed with Numbers; they are used to that Purpose (as certainly they have been contrived for it): Yet the other Way must be well understood, becaufe by it we speak to one another about Numbers, and the Refult of the Operations performed with the former: I fhall therefore explain them both. 2. IF we reflect, that Number is infinite or indefinite, then we fee plainly that 'tis impoffible to have different fimple Expreffions, for all the poffible Variety; nor yet for a great Variety, without making the Practice of infuperable Difficulty: But 'tis happily contrived, that we have but a few fimple Names and Characters; and by the Combination of thefe, we can exprefs any other in the following Manner. 3. If we confider Unity as a Number, then we conceive nothing as the Beginning of Number, or the Term from whence all Numbers commence, and we reprefent it in Arithmetick by this Sign o called a Cypher; to this join Unity or One, its Sign is 1; to this add another Unit, and you have the Number called Two and 2: Continue ftill to join another Unit, and you have these Numbers that are the Result of that Encrease expreffed both Ways thus, viz. Three, 3. Four, 4. Five, 5. Six, 6. Seven, 7. Eight, 8. Nine, 9. Thefe are all the fimple Characters in Arithmetick, and being arbitrary Signs of our Ideas, there is no Reafon to be demanded. FOR the Expreffion of all other Numbers by the Combination of thefe fimple ones, I must explain both the Ways diftinctly by themselves. 4. FOR the Expreffion of all Numbers above Nine, 9. by means of the former fimple Characters (which are ordinarly called the Digits, because there are as many of them as Fingers on one's Hands.) You must first know, that if Unity be added to Nine, the Sum is the Number TEN. And now by a very plain and eafy Artifice, we can, by the Combination of the former fimple Signs represent any poffible Number, by this general Rule, viz. when any Number or Combination of Digits (whofe Signification when they ftand alone A 2 we |