Otherwise, If you bid him add all his Products together in one Sum, and give you the Sum of them, then do you add all your Products together, and by that divide his Sum, the Quotient will give the Number thought. So 252 First SFirst Third 6804 The Sum 7812 Third 7 21 2189 217 Now if you divide 7812 (his Product) by 217 ( your Product), the Quotient will be 36, as before. upon VI. If feveral perfons, three, four, five, &c. should each of them think a feveral Number (under Ten), to tell what Number each person thought upon. BED ID the first perfon double the Number he thought, and add 5 to it, which Sum multiply by 5, and to it add 10; if from this Number you privately fubftract (always) 35, the firft figure of this remainder towards the left hand, fhall be the Number that the first Party thought upon. Let one person think The double thereof is To which add 5, it makes Example, This 23 multiplied by 5, makes From which 125, fubtract And the remainder is 90; from which the Cypher towards the right hand being omitted, the remainder is 9, the number thought. Thus 9 18 23 115 125 35 may you do if One perfon only think a number; but if Two, Three, or Four persons think feverally, it will be much the fame. Example, Suppofe there were three perfons, A, B, and C, and each of them fhould think these three numbers, A 4, B9, and C6. The Number which A thought is Which doubled, makes 48 To which 5 added, it makes Which multiplied by 5, produceth To which add to, and it makes To this Product bid B privately add the Number he thought It makes Which bid him multiply by 10, it makes To this Product let Cprivately add his Number thought, viz. It makes Which multiplied by 10, produceth This being done, bid them give you this laft Product 8460, from which do you privately fubftract 3500, and there will remain 4960, the Cypher being omitted, there will remain these three Digits, 4, 9, and 6, for the Numbers that A B and C feverally thought upon. Note, That if one perfon only Think, then (as before) fubftra&t from the laft Product only 35; if two perfons, fubftract 350; if three, 3500, &c. -VII. There lies in a heap 132 (or any other number which you know) of Counters, or other Pieces of Money, or any other things whatsoever: Then if Three perfons take each of them a certain number of Pieces out of the, heap, unknown to you, to know how many Pieces each party took. L ET the number of Counters in the heap be 132, and let the three perfons be A, B, and C : Bid one of the Parties, as A, take from the heap 4, 8, 12, 16, 32, &c. or any other number that may be divided by 4, and keep them in his hand: Then bid B for every four Pieces that A took, let him take 7 Pieces: And bid C for every four Pieces that A took, take 13 Pieces: Which when they have done, tell what number of Counters are left, and fubftract them from the Number you laid down, and the remainder will be the Sum of Counters which all of them have in their hands, which number do you privately divide by 3, the Quotient shall be double to the Number which A took. Example, Let the Number of Counters which you laid down be 132. Then fuppofe that A took out 16, that is, four times 4, then B muft take out 7 for every four that A took, which is four times 7, or 28; and C muft take out 13 for every four that A took, and that is 4 times 13, or 52. This done, tell how many Counters there are left upon the Table, which you find to be 36; this fubftracted from the whole heap, which was 132, there will remain 96; and fo many Pieces have A B and C in their hands; this Number 96 divide by 3, and the Quotient will be 32, the half of which is 16, and fo many Pieces did A take; then B must have taken 28, and C 52. VIII. If there be two Pieces of Money, as a Ninepence and a Shilling, or any other two Pieces (provided one be Even, and the other Odd), let any perfon take one of them in one hand, and the other in his other hand, to tell in which hand the Ninepence (or Odd Piece) is, and in which the Shilling (or Even Piece) is. BED D the Party that he double the Piece that he hath in his Left hand, and triple the Piece which he hath in his Right hand; then bid him add the two Numbers together, and ask him whether it be Even or Odd; if it be Even, the Shilling is in the Right hand; but if it be Odd, the Ninepence is in the Right hand. Or, If you bid the Party double the Piece which he hath in his Right ħand, and triple that which he hath in his Left, and if the Number be Even, the Odd Piece is in the Right hand, and the Even in the Left. IX. If IX. If any Number of Counters (fuppofe 9) or other Pieces of Money, Stones, or the like, be laid in a Row, to tell what Number (not exceeding the number of Pieces) any one thinketh upon. LAY Nine Counters on a row, (as in the Figure.) Then do you (privately) call that which lieth next your Left hand One, the next Two, the next Three, &c. then that towards your Right hand will be 9. Then bid any perfon think any number, not exceeding Nine, and then bid him lay his finger upon any of the Pieces; then (you knowing privately which Piece it is, whether the first, fecond, third, or any other) add 9 privately to the Number of the Piece he laid his finger upon. Thus if the Party fhould think 2, and he should lay his finger upon the Counter A, which is the fixth, to this 6 add 9 ) the Number of Pie ces in all) and it makes 15; then bid the Party count from the Number which he thought upon (beginning at the Counter A) backwards, till he make his number thought on 15, which Number 15 will end at the Counter B, which is the second Counter, and denotes the Number he thought on, to be 2. The like may be done with any Number more or less than 9; only remember, that as now you added 9, you must always add the Number of the Counters or Pieces you laid down, whether 6, 10, 15, &c. A Plain and Eafie Method of Extracting the Square Root of any Number (how great soever ) without the help of Multiplication or Divifion. TH HIS Method of Extracting of the Square Root, I cannot but attribute to Sir Sam. Moreland, Baronet, altho it doth not much differ from the manner of Extraction by Nepier's Bones: For as those Rods which my Lord Nepier (the Inventor of them) calls Rabdologia, Sir Samuel doth reduce the Pofition of them into feveral Tables, which he calls Tariffa's. The manner of making or preparing whereof Ishall here fhew: How How to make a Tariffa for One or more Figures, that is, for any Digit, or mixt Number. Example: Suppofe I would make a Tariffa for the Digit Number As in Table (or Tariffa) I. 4. First, Set down the Nine Digits orderly, in a Column towards the left hand, and in Figures bigger than ordinary, that they may take up the space of two lines of fmaller Figures. Secondly, Againft the Digit 1, more to the right hand, and at the top or upper part of the line fet 4, the Figure for which the Tariffa is to be made. Thirdly, Double 4, and it makes 8, which fet under 4, against the Digit 2, and at the upper part of the Line. Fourthly, Add 4 to 8, and they make 12, which fet against the Digit 3: Alfo to 12 add 4, it makes 16, which fet against the Digit 4; to which add 4, and it makes 20, which fet against the Digit 5, and fo continually adding 4, you fhall have 36 to ftand against the Digit 9 ; for 4 times 9 is 36. I Fifthly, In a Column yet one place forward towards the Right hand, fet the Squares of the feveral Digits; as against 1 fet i, against 2 fet 4, againft 3 fet 9, against 4 fet 16, and against 9 you will have 81: And thus is your First Tariffa for the fingle Figure 4 finished. Now to make a Tariffa for two or more Figures, the fame method is ftill to be observed. As for Example: Suppose I would make a Tariffa for 46. Firft, In a Column towards the Left hand fet down the Nine Digits orderly, and in a bigger Figure. Secondly, Set your Number to be Tariffed 46, againft 1, as in the fecond Tariffa. Thirdly, Double 46, and it makes 92, which fet against 2; then add 92 to 46, and they make 138, which fet against 3; and fo by the continual addition of 46, you fhall find 414 to ftand against the Digit 9, for 9 times 46 is 414. Laftly, In a Column yet more to the Right hand, fet the Squares of the Nine Digits; as againft i fet 1; against 2 fet 4; against 3 fet 9; against 4 set 16, as in the firft Tariffa. I And in the fame manner may you make a Tariffa for any number, how great foever by the directions of the two foregoing Examples; and as is done for the number 468 in the Third Tariffa; and for 4684 in Fourth Tariffa; &c. Digits. L How to Extract the Square Root of any Number. ET 5.48777476 be a Number given, and let the Square First fet down the Number, and make a Prick over the first figure thereof towards the right hand, and fo over every second figure, as is ufual in Vulgar Arithmetick, and then will the Number given being fo Pointed, ftand thus, 548777476 And there being five Points, it fhews that the Root thereof will confift of Five Figures, whereof to begin your Extraction, find the nearest Square Number to 5, the figure under the last Point, and that is 4, the Root whereof is 2, put 2 into the Quotient, and fet the Square thereof 4, under 5, and fubftracting 4 from 5, there will remain 1, to which I bring down the two figures belonging to the Second Point, namely 4 and 8, and then it will be 148, and the work will stand thus, 5 4 8 7 7 7 476 (2 4 148 Then double 2, the figure in the Quotient, and it makes 4; for which figure 4 make a Tariffa, as is before taught, (as the First Tariffa is) and ia Squares of the Digits. |