[Page] [Page] The Description and Use OF TWO ARITHMETICK INSTRUMENTS. TOGETHER With a Short Treatise, explaining and Demonstrating the Ordinary Operations of ARITHMETICK. As likewise, A Perpetual ALMANACK, And several Useful TABLES.

Presented to His Most Excellent Majesty CHARLES II. King of Great Britain, France, and Ireland, &c.

By S. MORLAND.

LONDON, Printed, and are to be Sold by Moses Pitt at the White-Hart in Little-Britain, 1673.

[...]

A New, and most useful INSTRUMENT FOR ADDITION AND SUBSTRACTION OF Pounds, Shillings, Pence, and Farthings; Without charging the Memory, disturb­ing the Mind, or exposing the Opera­tor to any uncertainty; Which no Me­thod hitherto published, can justly pre­tend to.

Invented and Presented to His most Ex­cellent Majesty CHARLES II. King of Great Britain, France, and Ireland, &c, 1666.

By S. MORLAND.

And by the importunity of his very good friends, made publick 1672.

# A[Page 1]TO Set or Dispose THE INSTRUMENT FOR OPERATION: Which Instrument must be held in the Left hand, and the Index or blew Pin in the Right hand, as men usually hold a Pen, but something more upright or perpendicular.

IN those five upper Plates, (in every of which there are 10 small holes) if any of the Digits (suppose 3) ap­pear in any of the Windows, put the point of the Index into the hole over against that Figure in the Margent, which is the Complement to 10. (namely 7.) And then turn it under the Window, where (0) will appear. This done, put the point of the In­dex into the (0) of the small Plate above it; and then, if you would prepare the Instrument for Addition, turn about that small Plate, till 0 stand just under that Line or mark in the upper Plate which is on the Right­hand, thus 11/0 as in Fig. A. But if for Substraction, then turn 0 on under the Left-hand line or mark, thus 11/0, as in Fig. D.

After the very same manner are all the lower plates of Shillings, Pence, and Farthings to be disposed; Re­membring onely that in the place of Shillings, where (20) is the number, the Gomplement to (3) is (17) And in Pence, the Complement to (3) is (9) And in Farthings, the Complement to (3) is (1) And [Page]

[Page 2] thus are all the Plates fitted for Operation as in Fig. A.

This is likewise to be observed before the Instru­ment is set: If any of the Plates with Holes stiek, and will not turn about forwards, it is because neither (0) nor any Figure of the small Plate above it is under the Right-hand-line or mark. And if they will not turn backwards, it is because neither O, nor any Figure of the small Plate above it is under the Left-hand-line or mark. And lastly, if the small Plate above will not move, it is because no Hole in the lower Plate is di­rectly under the Window.

Let the Sums to be added together, be these.

 l. sh. d. f. 7 14 03 2 48 11 10 3 252 17 11 1 4337 8 2 1 4646 12 3 3

First I set on 7 l. that is, I put the point of the Index into the hole of the place of Ʋnites of Pounds, which is over against (7) in the Margent, and turn it under the Window. After that, I turn on (14) in the place of Shillings; (3) in the place of Pence; and (1) in the place of Farthings.

Again, as to the 2 Summe, I set on (48) thus, namely, (4) in the place of Tens, and (8) in the place of Ʋnites: As likewise 11 in the place of Shillings; (10) in the place of Pence; and (1) in the place of Farthings And having set on all the Summes in the manner and method aforesaid, they will appear on the Instrument as in Fig. B.

In fine, to perfect the Operation, I begin with the place of Farthings, and because the Figure (1) of the [Page]

[Page 3] Plates over head stands between the two Lines or Marks, therefore I turn on (1) in the place of Pence. And because (2) is between the two Lines in the place of Pence, I turn on (2) in the place of Sbillings: And because (2) is between the two Lines in the place of Shillings, I turn on (2) in the place of Ʋnites of Pounds. Thus, for (2) between the two Lines in the place of Ʋnites of Pounds, I turn on 2 in the place of Tens. And for (1) in the place of Tens, I turn on (1) in the place of Hundreds. Which done, the true Summe or Aggregat of all the Summes, appears in the several Windows, viz. 4646 l.—12 s.—3 d.—3 f. As in Fig. C. The same thing may be performed by ad­ding together, First the Farthings apart, then the Pence, next the Shillings, and lastly the Pounds.

If peradventure the Summes are so numerous, as that by often turning of the Plates with holes, the small Plates above them are like to be over-charged, That is, if the Figure (7) (8) or (9) of those small Plates come between the 2 Lines or Marks, then I dis­charge them, by turning on the same numbers in the next places: For example, if the Figure (7) of the small Plate above the place of Farthings, be under the Line, I turn on (7) in the place of Pence. If the Fi­gure (7) of the small Plate above the Pence be between the two Lines, I turn on (7) in the place of Shillings, and so to the end. And lastly, I turn O of each small Plate under its proper Line, And hen proceed (without setting the lower Plates to (0) or at all altering them otherwise than aforesaid) to set on the remaining Summes, to a Million of Pounds. (Or if it be desired, the Instrument may be made for a far greater Summe.) But the surest way, is to divide a long Page into two or three parts, and so to work them distinct­ly.

## [Page 4] The Operation of Substraction.

Having prepared the Instrument for Substraction as is before directed, suppose I would from 327 l. - 12 s. - 07 d. deduct 39 l. - 14 s. - 3 d.

First, I set on 327 l. - 12 s. 7 d. in their proper places. Then I begin with 3 d. of the Summe to be deducted, and set it backward, That is, I put the point of the Index into the hole under the Window of Pence, and turn it till it stand over against (3) in the Margent, which will leave (4) in that Window. Thus I set 14 s. back­wards in the place of Shillings; And thus (39) in the Ʋnites and Tens of Pounds. And having so done, I ob­serve if any (0) of the small Plates over head be remo­ved from under its Line (as in Fig. D. in the place of Shillings, I find the small Plate over head moved) I put the point of the Index into that hole of the next place, that is under the Window, and turn it backwards under (1) in the Margent. After the same manner, because the small Plate over the place of Ʋnites of Pounds is re­moved out of its place, I set back for it (1) in the place of Tens; And so likewise, because the small Plate over the place of Tens is removed, I set back (1) in the place of Hundreds. Which done, the remaining Summe, viz. 287 l. - 18 s. - 04 d. is found in the re­spective Windows. As in Fig. E.

# [Page 5]The Description and Use of an Additional Wheel, applicable to this New INSTRƲMENT, for all those who shall desire it, which renders it useful, beyond exception, for the longest Accompts, without either dividing the Page, or setting ☉ of any of the small Plates under their Lines a second time.

BƲt forasmuch as it may and will often happen, in the adding up long Pages of Accompt-Books, That the small Plates will be over-charged (and that more than once in a Page) which may be something trouble some to the Ope­rator; Therefore I have contrived an Additional Wheel, as in Fig. F. so that no one or two Pages of any Accompt-Book whatsoever can overcharge it: all the difference is on­ly this, That what Figure soever of this Wheel is found op­posite to the Line (or mark) above it, it must be set on (not in the next place, but) in the next place save one, in the manner following: That is to say, The Figure of the small Wheel above the Farthings, opposite to the Line above it, must be turned off in the place of Shillings. The Figure of the small Wheel above the Pence, opposite to the Line above it, must be turned off in the place of Unites of Pounds. And if any Figure of the small Wheel behalf a division beyond the said Line, there must be 10 turned off for it in the place of Shillings, (which is the great­est intrigue in the whole Operation, For,) The Figure of the small Wheel above the place of Shillings opposite to the Line, must be turned off in the place of Tens of Pounds. The Fi­gure of the small Wheel opposite to the Line, in the place of Unites of Pounds, must be turned off in the place of Hun­dreds. Lastly, the Figure of the small Wheel opposite to the Line, above the place of Tens of Pounds, must be turn­ed off in the place of Thousands; and so to the end, as in Fig. G. So that by the addition of this small Wheel, a [Page]

[Page 6] place more is gained, and the Instrument, if need were, would add up an Accompt of Ten Millions of Pounds, by the Example, compared with Fig. G. will more evidently appear.

EXAMPLE.

 l. s. d. q. 76534 13 03 3 76534 10 11 2 85637 14 05 1 93792 17 10 2 74379 09 08 3 85466 19 10 2 72954 10 04 2 61117 16 08 3 52252 16 06 3 65577 15 05 3 42573 15 02 3 72576 19 09 2 69955 17 11 3 72777 13 09 1 85855 04 10 3 98888 14 06 2 97744 16 07 2 45757 12 04 3 73879 16 11 3 64549 19 08 2 97872 14 06 1 85678 10 10 1 76644 08 06 3 56279 18 03 2 45772 15 05 3 56572 17 10 2 67699 14 11 2 71578 18 08 2 89979 19 10 3 97979 01 06 3 95878 08 11 2 86788 10 09 1 99678 13 10 2 89485 17 06 2 47632 16 03 1 57416 15 09 3 62517 14 07 2 74528 13 11 2 43215 12 05 2 67742 11 08 3 35418 09 04 1 92261 01 09 3 44415 12 06 3 43324 14 03 1 37338 02 11 2 34512 19 04 2 56735 17 02 3 3283761 11 06 1

# [Page] Machina Nova CYCLOLOGICA Pro Multiplicatione. OR, A new Multiplying-INSTRUMENT: Invented, and humbly present­ed to the Kings most Ex­cellent MAJESTY CHARLES II.

THe Fabrick of this Instrument be­ing truly represented in Perspect­ive, there will be no need of any large Description of it.

The 5 upper Circles whereon are the Figures 0/9 ⅛ 2/7 3/6 ⅘ re­present 5 moveable Plates which lie always ready to be taken off the 5 Semi-circular Pinions (or Centers) S, T, Ʋ, W, X, and to be set on any of, or all the Centers (or Semicircular Pinions) a, e, m, n, o, p, interchangeably. And underneath each of those upper [Page]

[Page 9] Plates aforesaid,, are placed 5 other Plates of the same dimensions, and with the same Figures graved on both sides; so that of these upper Plates there are 30, that is to say, Six of each kind, besides 5 other Plates un­der the place marked Q / QQ which serve for extracting the Square, Cube, and Square-Square-Roots, of which in its place.

E. F. is a Line divided into 9 equal Parts, on which runs a small Black Pin or Index forwards or backwards at pleasure, being turned about by G. H. which is like to the Key of a Watch.

Lastly, PQ is a Plate opening with hinges, and shut­ting down upon the lower Circular Plates, and a little Bolt at R. locking it down, which Plate has 6 square holes (or foramina) through which are discovered such Figures onely, as are necessary for the operation.

K, L, M, is a long Rack on the backside of the In­strument, by which all the Plates are turned about on the Pinions a, e, m, n, e, The meaning of all which is this. The Instrument is to be supposed as it is represent­ed in Figure A save onely that there are as yet no Cir­cular Plates taken off the Semi-circular Pinions S, T, Ʋ, W, X, and that the number given to be multiplied is (1734.) Then one of the Plates marked ⅘ is first to be taken off the Pinion x and set on the lower Semi-circu­lar Pinion (r) which is the furthest on the right hand. Afterwards one of the Plates marked 3/6 is to be taken off the Pinion W, and set on the Pinion (o) next to the other.

Thirdly, one of the Plates marked 7/2 is to be taken off the Pinion V and set on the Pinion (n.)

Lastly, one of the Plates marked ⅛ is to be taken off the Pinion T, and set on the Pinion (m) Which done, the Plate P Q. is to be shut down, and then will appear the number 1734 through the holes, as in Fig. B and this is all the trouble of the Instrument in this or any operation.

[Page 10] The Figure B represents the very same posture of the Instrument in the foregoing Figure A save one­ly that the Plate P Q is now shut down and lockt upon the four lower circular Plates marked on the former Figure A with the large Figures ⅛ 7/2 3/6 ⅘ which was in effect, a preparation for the Multiplication of the Number (1234) which Number now ap­pears through the small holes or foramina of the said Plate P Q hiding all the other Figures from the Eye of the Operator. In which particular this In­strument far surpasses the Lord Napiers Lamina or Bones, which expose a great number of Figures to the Eye at the same instant of time (as well those that are not useful as those that are) besides that all the Figures in those Bones are placed Diagonally, which does very much strain and force the Eye of the Operator; Whereas in this they lie all in a straight Line, and as distinctly as can be desired.

The Instrument being disposed as aforesaid, sup­pose the given Number 1734, be to be multiplied by 24. I first set it down in Writing thus [...] then first I turn then Hand or Key G H till the Index point to 4 in the Line E F so have I the first Product (6936) given me in the Windows or Foramina of the Plate P Q, as in Fig. C which I set down thus

This done, I turn the Hand or Key back­ward, [...] till the Index point to the Figure (2) in the Line E F and in the Holes or Foramina of the Plate P Q (in Figure D.)

So have I given me the

[Page 11] The Second Product, viz. 31468. which I set down under the first Product, and add all of them as in the ordinary Method of Multipli­cation, or else by the help of Napiers Bones.

[...]

And after this manner may any Number be Multi­plied by the help of this Instrument, which is capa­ble of being made for any Number of places.

This Instrument alone is also of excellent use in Di­vision; as likewise in extracting the Square, Cube, and Square-Square-Roots, for which the 6 Plates at the end of the Instrument on the right hand are extreme ready and serviceable, and are to be used after the very same manner as are the Lord Napiers Bones, and therefore need no further explication.

But if any person have the Curiosity, and is willing to goe to the Expence; the Adding Instrument being joyned to the Multiplying Instrument, performs Addition, Subtraction, Multiplication, and Division; as likewise the Extraction of the Square, and Cube-Roots, &c. without the help of Pen and Ink, or exposing the Operator to any difficulty or uncertainty.

SOLI DEO GLORIA.

# A CAUTION To all who desire to make Use of either of these INSTRUMENTS.

IF any person desire to have either of these Instruments exactly made, and so as it may be serviceable for many years. He may bespeak it of Mr. Humphry Adamson, living at pressent at the House of Jonas Moor, Esq in the Tower, who is the onely Work­man that ever as yet could be found by the Author to perform the said In­strument, with that exactness that is absolutely necessary for such Operati­ons.

FINIS.

# A Short DISCOURSE Concerning the ANTIENT, & COMMON WAY OF NUMBRING.

BUt for the better understanding of these Arithmetical Instruments, I shall en­deavor so to explain and demonstrate the reason of the Operations of Addi­tion, Subtraction, Multiplication, Division, and Extraction of the Square, and Cube-Roots, as to render them plain and obvious to the meanest capacities.

The way of Numbring in use with the Antients, was by the Letters of their respective Alphabets; For example: with the Romans C signified 100. D 500. M 1000. ↃMC 10000. CC.I.ↃↃ 20000. CqↃ 100000. &c.

So among the Grecians, Α or α. signified 1.▪ Β or β 2. Τ or τ. 3. Ι ι. 10. Ρ ρ. 100. α, 1000. δ▪ 4000. ζ. 7000. &c.

But note that the most common way of expressing the value of Numbers, is by the Arabick Notes or Charact­ers called Ziphers, by the Hebrews, Sephers; and by Us, Cyphers; and they are ten, viz. 1, 2, 3, 4, 5, 6, 7, 8, 9, 0.

The way of placing those Characters, is from the right hand to the left, after the manner of the Hebrews, in their Writings, as Gemma Frisius and others observe.

[Page 14] The Progression of them is Decimal, or by Tens; for every place to the left, is ten times the value of the next place to the right, as in the following Table.

 The Third Period The Second Period The first Period Hund. of Millions Tens of Millions Unites of Millions Hund. of Thous. Tens of Thous. Unites of Thous. Hund. of Unites Tens of Unites Unites of Unites C X I C X I C X I M M M M M m u u u 9 8 6 1 5 7 4 3 2 Nine H. Eighty six Milli. One H. fifty seven Thous. Four H. thirty two Unites.

In this Table, the first Period is of Ʋnites, the second of Thousands, the third of Millions, Those who would proceed to more Periods then what are here set down, may give them what ap­pellations they please, as Billons, Triltions, Quadrillians, Quintillians, &c. as Tacquet and others have done before us.

But the best and surest way of placing and distinguishing great Numbers, is to put a Comma, or other mark of distinction betwixt every three of them, thus, 12, 345, 769.

# CHAP. 1. The Precept for ADDITION of Integers in Plain Numbers.

HAving placed the Unites of the respective Pro­gressions in Ranks and Files; then begin and add together the Unites of the right-hand-File, setting down the sum underneath, if it be under 10. but if just ten, set down 0. and carry 1. to the next place; and if above 10. set down the excess in the first place, for every 10. an Unite.

Examples.

[...]

# CHAP. II. The Precept for SUBTRACTION of In­tegers in Plain Numbers.

HAving placed the less number under the greater, according to the respective places or Progressions, begin at the right hand, Substracting the lower Figure out of that above it, and setting the Remainder under­neath. But if the Figure chance to be the less, (as oft it falls out) then there must be an Unite borrowed, and brought from the next Progression, to supply the defect, which must be again repaid, by adding an Unite to the next lower Figure on the left, which is the same [Page 16] thing as if the Figure above it were diminished by an Unite. And for a proof of the operation, the number Subtracted, and the remainde, must evermore equal that out of which it was Subtracted.

For out of a number A C let a less number A B be de­ducted, Then by the Hypothesis, A B together with the remainder B C are equal to the whole number A C For­asmuch as The parts united, are always equal to the whole.

[...]

# CHAP III. The Precept for MLTIPLICATION of Integers in Plain Numbers.

HAving placed the Numbers one under the other as in Addition, Multiply the last right-hand Figure of the Multiplicand, by the last right-hand Figure of the Multiplicator, and set the Product (if less then 10) un­derneath; but if greater, carry the excess (that is for every 10, an Unite) to the next place. And if the Mul­tiplicator have more places then one, set down the first Figure of each respective Product under that Figure of the Multiplicator by which it was made, and so on to the left: observing Ranks and Fyles.

[Page 71] For example, 7 times 6 is 42, that is setting down 2 in the place of Unites and carrying 4 in the mind to the next place. Again 7 times 2 is 14, and 4 that was car­ried in mind makes 18, that is setting down 8, and carry­ing 1 to the next. Again 7 times 4 is 28, and 1 is 29.

[...]

And so in the second Pro­duct twice 6 is 12, that is 2 carrying 1 to the next, and so on to the end.

The reason of which operation is plain by the follow­ing Table.

 CM XM M C X U [...]. 4 2 6 Multiplicand. 3 2 7 Multiplicator. For 7 times 6 is 42; and 7 times 20 is 140; and 7 times 400 is 2800, or two thousand eight hundred. 4 2 First Product 1 4 0 Second Prod. 2 8 0 0 Third Product Again, 20 times 6 is 120; and 20 times 20 is 400; and 20 times 400 is 8000 or Eight thousand. 1 2 0 Fourth Prod. 4 0 0 Fifth Product 8 0 0 0 Sixth Product Thirdly, 300 times 6 is 1800; 300 ttimes 20 is 6000; & 300 times 400 is 120000, or One hundred and twenty thousand. 1 8 0 0 Seventh Prod. 6 0 0 0 Eighth Prod. 1 2 0 0 0 0 Ninth Prod. The sum total of which Products amounts to One Hun. thirty nine Th. three Hund. and thirty two Unites. 1 3 9 3 3 2 Summe total of all nine Products.

# CHAP: IV. The Precept for DIVISION in Plain Numbers.

DIVISION, is in effect nothing else, but the de­ducting of a less number as oft as may be out of a Greater, and so finding at last the number, by whose U­nites that less number being repeated, makes a number equal to the Greater.

Now the greater of these numbers is Vulgarly called the Dividend, the less the Divisor, and the last the Quo­tus or Quotient.

The method of this Operation is thus;

1. Set the Figures of the Divisor under an equal num­ber of Figures of the Dividend on the left hand, if those Figures of the Dividend be of greater, or at least of equal value with those of the Divisor: Otherwise you must place the first Figure of the Divisor under the second Fi­gure of the Dividend. And having set the Divisor right, put pricks over the Figures of the Dividend, from the Unite place of the Divisor, inclusivè. And the number of pricks denote the number of places in the Quotient.

2. You must evermore prepare such a Tariffa (or Ta­ble of Multiplication) for the Divisor, as is here set down on one side of the Operation, and is of excellent use, making the work ten times more easie and certain.

3. You must find by the Tariffa how many times the Divisor is found in those Figures of the Dividend under which they are placed, and the answer to that, is the first Figure of the Quotient; by which you have multiplied the Divisor in the Tariffa, then deduct the product out of those upper Figures of the Dividend, and what remains must be considered in the next operation, if there be more pla­ces then one in the Divisor.

4. The next Figure of the Dividend must be taken down and set next to the Remainder, if there be any. And the Divisor must be again set under it, if the value [Page 20] of the upper Figures be sufficient; if not, there must a Null or (0) be set in the Quotient, and then the next Figure of the Dividend taken down, and the very same Operation repeated, till the work be at an end. But one Example in things of this nature clearly and distinctly set down, is better then a thousand verbal directions.

Let the Dividend be that Number, which was last found by Multiplying (426) by (327)

That is to say, Let the Dividend be And the Divisor be [...]

Having pointed the Dividend, and pla­ced the Divisor un­der (1393;) look for 1393 (or the nearest number to it) in the Tariffa, which is 1308. wherefore I set that down; and sub­tracting it from 1393, there remains 85; then (having set down 4 for the first Figure of the Quotient) I take down the next Figure, or Cypher of the Dividend, viz. (0) which makes it (850) In this (327) the Divisor by the aforesaid method is found twice; wherefore I set (2) in the Quotient, and then deduct the Product, viz. (654) out of it, and there remains (196) to which in the last place, I take down (2) the last Figure of the Dividend, and make it (1962) in which (327) is found 6 times, and so the work is at end.

[Page 21] The reason of this Operation is plain in the subsequent Table.

 C M X M M C X U 1 3 9 3 0 2 Dividend. 3 2 7 Divisor, which is advanced 3 places, and is not now 327. but 32700. Quot. 400. 1 3 0 8 This Product is not 1308, but 130800, which is 4 times 32700, that is 420 times 327, which is the Quotient in the Margent. 8 5 That is 32700 being deducted out of 130800, the remainder is 8500, and so is ended the first Operation. 8 5 0 Here begins a second Operati­on. And this is not 850, but 8500. 3 2 7 This is not 327 but 3270, which being doubled makes 6540, which in effect is 20 times 327, as is exprest by the Quotient in the Margent. Quot. 20. 6 5 4 1 9 6 That is, 6540, being deducted out of 8500, there remains 1960, and so ends the second Operation. 1 9 6 2 3 2 7 Here 327 is brought down to its own value again. Quot. 6. 1 9 6 2 That is 6 times 327 is just e­qual to 1962, and so the work is at end. 0 0 0 0

[Page 22] But now if this Dividend had been greater by 20 Unites, that is, if it had been (139322) the work had been the same, and the Quotient had been the same num­ber of Integers, but there had been found remaining a broken part of Fraction of 20. which must have been be set thus 20/327

# CHAP V. Notation of FRACTIONS.

A Broken Number, (otherwise called a Fraction) is part of an Integer; For example, A Foot in length contains 12 Inches. One pound 20 shillings, One shilling, 12 pence; One penny, 4 Farthings, &c.

The Parts of a Fraction.

A Fraction consists of 2 parts, The Numerator and the Denominator, which are placed one above the other, and separated by a little line; For example, If I would ex­press three quarters of a yard, it must be set thus ¾ Numerator Denominator.

F. Inch.

Three Foot and nine Inches, thus. [...]

Or if a Foot be divided in­to an hundred parts, thus [...]

Or if a Foot be divided onely into four parts, thus [...]

And so are all other broken Parts or Fractions exprest of what kind soever.

# CHAP. VI. The Reason of Translating Fractions from one Denomination to another, as likewise of reducing them to their least Tearms, and the truth of the Operations demonstrated from several Propositions of Euclids Ele­ments.

HAving thus explained the nature of Fractions, I shall in the next place proceed to demonstrate the Reason of Translating Fractions from one Denomi­nation to another, as likewise the Reason of depressing or reducing them to their lowest and least Terms. All which is necessary to be known by those who desire to be Masters of Fractions.

First of Translating Fractions from one Denominations to another.

## Eucl. l. 7. Theor. 16. Prop. 18. [...], &c.

If two numbers multiplying any number, produce other numbers, the numbers produced of them, shall be in the same proportion that the numbers multiplying are, vid. Fig. E.

For the applying of this Proposition to the matter in hand, I lay before me these two Fractions, namely ⅔ and ¾ as in Fig. F. And first I single out ⅔ and Multiply them by (4) the Denominator of the next Fraction and they become 8/12 for 4 times 2 is (8) and 4 times 3 is (12) as may be seen in Fig. G.

[Page 24] This done, I say, that by thenfore-mentioned Propo­sition, this Multiplication of the two numbers (2) and (3) by a 3d number (4) has altered no proportion, but the products have the same proportion one to another, as the numbers (2) and (3) have, by which they are multipli­ed, that is to say, as (2) is to (3) so is (8) to (12).

Again, I single out the other Fraction, viz: ¾ and I multiply these two by a third Number, namely by (3) the first Denominator. And the Products are 9/12 as in Fig. H. Neither hath this Multiplication altered any proportion, by the same reason with the former. For as (3) is to (4) so is (9) to (12). By which means I have two new Fractions, viz. 8/12 and 9/12 which are the same in effect with the first two, viz. ⅔ and ¾. And whereas they were before of different Denominations, they are now brought under one and the same Denomination.

And thus is that 18th Prop. of the 7th Book of Euclid, the true ground and reason of altering and translating Fractions from one Denomination to another; And from divers and different Denominations, to one and the same. Which Proposition being once throughly digested, and imprinted in the mind and memory, all other Operati­ons which relate to Fractions, (as Adding, Substracting, Multiplying, and Dividing them, as also extracting their Square and Cube Roots) will admit of very few or no difficulties.

Secondly, The way of reducing Fractions to their least Terms.

But forasmuch as it often happens in the multiplying and translating of Fractions that the swell into too great numbers, which are not so tractable as smaller numbers are. It will be proper in the next place, to shew the way of reducing them to their least terms, either before, or after they are thus multiplied or translated, as the practitioner shall see occasion. For the better ef­fecting of which, he is desired to consider some few Definitions and Problems of the 7th Book of Euclid's Ele­ments,

## Eucl. l. 7. Defin. XI. [...], &c.

A prime number is that which is measured onely by an Unite.

That is to say 2, 5, 7, 11, 13, &c are prime numbers, because neither of them can possibly be divided into e­qual parts by any thing less then an Unite.

## Defin: XII. [...], &c.

Numbers prime the one to the other, are such as only an Unity doth measure, being their common measure.

And such are (9) and (14) for these two numbers can­not be divided into less then Unite parts, so as that which measures one, may measure the other.

For though (3) will measure [...] 9, that is to say, it is found 3 [...] times in (9) yet it will not measure (14) that is, after it has been found as many times as it can be found in (14) there will be two odd Unites left.

Again though (2) do measure (14), that is to say, is found just 7 times in (14) yet it cannot be found any certain number of times in (9) but there will be an odd Unite or Unites left.

## Defin. XIII. [...], &c.

A composed number is that which some certain num­ber measureth.

So is (15) a composed number, because (3) multiply­ed by (5) makes (15). And so is (20) a composed number; because 4 multiplyed by (5) makes a Product of (20).

## Defin. XIV. [...], &c.

Numbers composed, one to the other, are they, which some number, being a common measure to them both, doth measure.

And thus are (8) and (12) composed Numbers one to another, because there is a certain number, viz. (4) which being repeated, or added to it self a certain num­ber of times, composes both the one and the other of these numbers. That is to say, (4) being repeated (or added to it self) twice, composes (8) the one of these numbers, and the same (4) being repeated (or added to it self) 3 times, composes (12) the other of these numbers. And this (4) is called the common measure of (8) and 12. as in the following Table.

 Two composed Numbers 8 and 12 Divisors of these 2 composed Numbers. 2 3 Their common measure. 4 Three composed Numbers 6 & 8 & 12 Divisors of these 3 eomposed Numbers 3 4 6 Their common measure 2

## Probl. 1. Prop. 2. [...], &c.

Two Numbers being given (not prime the one to the other) to find out their greatest common Measure.

For example, let the Numbers given be (A=15) and (B=9)

I take the less Number [...] (B=6) from the greatest (A=15) as oft as I can, which is once: If nothing remain then (B=9) is the greatest common measure. If something? Let it be (C=6), and then I [Page 27] take (C=6) out of (B=9) as oft as I can, and if there remains nothing, then (C=6) is the greatest common measure; and if there remain something, let it be (D=3) then I take (D=3) twice out of (C=6) and because nothing remains, therefore (D=3) mea­sures (C=6) and therefore it measures (B=9) and al­so (A=15) And is likewise the greatest Number that measures B and A, that is to say their greatest common measure. And this will be in all other Numbers (not prime one to another) before I can come to an Unity.

## Probl. III. Prop. 35. [...], &c,

How many Numbers soever being given, to find the least Numbers, that have the same proportion with them.

Let the Numbers propounded be (A=6) (B=8) (C=12) either these are Prime Numbers, or else there is a certain Number which being repeated, (or added to it self) a certain Number of times, composes either of them, and is therefore their common measure.

If they be Prime Numbers, I have granted what I desire.

[...]If not, let then their greatest com­mon measure be D. And look how many times D is found in A, let E have so many Unites. And let F have as many Unites as D is found times in B. And G as many Unites as D is round times in C. That is, (D=2) multiplied by (E=3) produces (A=6). And (D=2) multiplied by (F=4) produ­ces (B=8). And (D=2) multiplied by (G=6) pro­duces (C=12).

Then E, F, and G, are the least Numbers that have the same proportion with the first Numbers, A, B, and C.

# CHAP. VII. Addition of FRACTIONS.

LEt the given Fractions to be added together be ⅔ and ¾.

First I reduce them by the 6 Chapter, to 8/12 and 9/1

Then I add 8 to 9, which makes it 17/12 or 1 5/12.

If they be more then two. First reduce two of them to one Denomination. And then the sum of those two, and the next, &c.

For example, Let the Fractions be ⅔.¾.⅘. First, ⅔ and ¾ make 17/12. Then 17/2 and ⅘ being added as be­fore, make 36/60 and 85/60. which make 121/60 or 2 1/60

# CHAP. VIII. Subtraction of FRACTIONS.

FRom ¾ are to be deducted ⅔. That is by the forego­ing Rules from 9/12 are to be deducted 8/12. There­fore there remains 1/12.

# CHAP. IX. Multiplication of FRACTIONS.

THis is onely to Multiply the Numerators one by an­other, for a New Numerator, and then the Denomi­nators one by another for a New Denominator.

[Page 29] For example. To multiply ⅔ by ¾ is to multiply 2 by 3. (6) for a new Numerator, and then 3 by 4 (12) for a new Denominator. This is ⅔ by ¾ give 6/12 or ½ for the Product.

But to make all more plain, and to present both the Operation and the Reason of it, to the Readers eye, at one and the same instant; let him consider the less Ob­long (6) in Fig. L. (made by multiplying the Numera­tor (2) by the Numerator (3) included in the greater Oblong (12) made by multiplying the Denominator (3) by the Denominator (4) then which nothing can possibly be more plain, or satisfactory.

# CHAP. X. Division of FRACTIONS.

THe Rule for Division of Fractions, is this,

Multiply the Denominator of the Divisor, by the Nu­merator of the Dividend, for a new Numerator; And the Numerator of the Divisor, by the Deno­minator of the Dividend, for a new Denominator. And the new Fraction is the Quotient.

For example, if ¾ be to be divided by ⅔

Then is ¾ the Dividend.

And ⅔ the Divisor.

• Then / 3 into 3 make a new Numerator, viz 9/8
• And 2/ into / 4 make a new Denominator, viz. 9/8

that is 9/8

That is to say ⅔ is contained in ¾ once, and one eighth part. 1 1/8

The Reason of this operation is this.

First, I turn the two Fractions ¾ and ⅔ into two new Fractions, viz. 9/12 8/12 (as before has been shown). Then, as if they were plain Numbers, I divide (9) by (8) and the Quotient is (1 1/8). That is to say, the Di­vidend (¾) is to the Divisor (⅔), as (9/12) to (8/12). But now (9) is the Product of the Divisors Denomina­tor Multiplied by the Dividends Numerator, and is the Numerator of the Quotient, And (8) is the Product of the Divisors Numerator Multiplyed by the Dividends Denominator, and is the Denominator of the Quotient; which is consonant to the foregoing Rule, and that which was to be demonstrated.

[Page 31] Thus if 6/12 be to be divided by ¾.

/4 into 6/ make a new Numerator 24 And 3/ into / 12 make a new De­nominator 36 viz. 24/36

Which being reduced to its least terms, is ⅔

# CHAP XI. Of Decimal FRACTIONS.

THese Decimal Fractions are of all Fractions the most natural: For the truth is, all plain Num­bers as they are exprest by the Arabick Notes of 1, 2, 3, &c. are nothing else in effect, but Decimal Fractions. For example, Let any Number be given (432) The last Figure (2) is really 2/10, and so are the last two Fi­gures (32), truly and properly 32/100. And all three are a Fraction of 432/1000. For as the very progression of these and all other plain Numbers is Decimal; that is, each Figure on the left hand, is ten times the value of the same number placed in the next place on the Right. Thus in (333) the last (3) on the left is ten times the value of the (3) next to it on the right; And the (3) in the middle, is ten times the last (3) on the right, and but the tenth part of that (3) on the left. And therefore all, or any of them may be pro libitu, either Fractions or Integers: If I would have them Integers, I set them down without any Line drawn under them; But if I would have the two last a Fraction, I put a se­parating Comma between them, thus, 3, 33, that is 3 Integers, and 33/100 for the Denominator is here to be un­derstood to be an Unite of the next place or Denomina­tion, as was before explained. And thus 5270 is an Integer or whole number, and the same number with a se­parating [Page 32] Comma, thus 5, 270 is an Integer (5) with a Fraction of 270/1000. Or the whole is a Fraction of 5270/10000.

# CHAP. XII. Of Addition and Substraction of Decimals.

THe Operations of Addition and Substraction in De­cimals, Integers, and Fractions, is the very same with that of plain Integers, only the careful setting the Unites of all the Integers in one File, and if there be any void places, they are to be imagined to be filled up with Cyphers.

[...]

Examples of Subtraction.

[...]

# CHAP. XIII. Multiplication of Decimals.

THis is likewise the very same Operation with that of Multiplying plain Integers, save onely, when [Page 33] all the work is ended, there must be as many places of Parts, or Fractions in the Product, as there were places both in the number multiplied, and in the number multi­plying. As in the following Example, there are two places cut off in the number multiplied, and one in the number multiplying; Therefore there are three places cut off in the Product. And in the second Example, be­cause there are three places cut off in the Multiplicand, and two in the Multiplicator: therefore there are five cut off in the Product.

• 1. Example. [...]
• 2 Example. [...]
• 3 Example. [...]
• 4 Example. [...]
• 5 Example. [...]
• 5 Example. [...]

In the 4 Example, because there are no Integers, the Product is 783/1000

In the 5 Example, because there do not arise but five places in the Product, viz., 01875 and yet by the Rule there ought to be six places cut off, therefore two Cy­phers [Page 34] must be prefixt to make up the number of six pla­ces, viz. 001875/1000000

# CHAP. XIV. Division of Decimal FRACTIONS.

1. IF the Dividend be greater then the Divisor, the Quo­tient will be either a whole number, or a mixt; but when the Dividend is less then the Divisor, the Quo­tient must be a Fraction.

2. Whatever the Diuidend be, if need require, there must be a competent number of Cyphers added to it, so make room for the Divisor to stand under it, and then it is no other but ordinary Division, as will appear by the following Example.

Dividend 172, 5

Divisor 3, 746

which the Dividend being supplied with Cyphers, stands thus

[...]

# CHAP. XV. The Vulgar Precept for extracting the Square Root or side of any Plain Number.

HAving first pointed the given Number, (suppose 625) that is to say, sett a prick or point over every other Figure, beginning with the last. Whereby I know that the Root has two places and no more. First, I enquire whether (6) the last pointed Figure or Figures, be a true square Number. If it be, then I set down the Root (which is easily found by the following Table, and indeed ought to be retained in the memory) some where on the right hand of the work, and so is the first Figure of the Squar-Root found. But if it be not a true Square Number, then I take the Root of the Square next to it, as in this case, I see (6) is not a true Square, I therefore take the Root of (4) which is the nearest to it, and make that the first Figure of the Root. And then Subtracting (4 the Square thereof) out of (6) I have left remaining (2) which I set underneath, and so is the first Operation ended.

[...]

This done, I take down the Figures of the next Square Number, viz. (25.) Then I set the double of the first Figure of the Root under the first Figure of the second Square Number; and I seek out a Digit, which being multiplied into it self, together with the double of the Quotient (or first Figure) may take a­way the remaining Figures of the given Number, viz. (225) or at least as mvch as may be; which Digit, by as king how many times (4) the double of the Root is sound in the Figures standing over it, as in the nature of a Dividend (namely 22) for I find that (4) will be found in (22) 5 times, and enough over and above to [Page 36] multiply that 5 into it self also. Therefore I set down (5) for the second Figure of the Root, and I also set the same (5) under the last Figure of the second Square; and then I multiply the double of the first Figure, augmented by that second Figure (5) that is, I multiply (45) by that (5) and if the Product be ei­ther equal to the Figures (225) standing above them, or so much less then (225) as the value of any Num­ber under (45) then the second Figure is rightly chosen otherwise the Figure next less must be taken. But in this example, the Product happens to be just equal And so the work is ended.

[...]

Now in case this (625) had been (655) which is a greater Number then (625) by 30 Unites; the Integers of the Square Root had still been (25;) onely there had been left (30) which had been the Numerator of a Fracti­on, whose Denominator must evermore be the double of the Root, augmented by an Unite, and then the Operation had been, as here you see.

[...]

# CHAP. XVI. The Reason and Demonstration of the Vulgar Operation of Extracting the Square-Root.

ANd after this very manner and method may the Square-Root of any Plain Number in Integers be [Page 37] extracted, though never so great; but that this and all other Operations of the same nature may be also performed with understanding, and satisfaction, it will be necessary to make some reflections upon the nature and genesis of a Square-Number, and in order there­unto, the Practitioner is desired to consider the fol­lowing Definition of a Square-Number.

## Eucl. l. 7. Defin. XVIII. [...], &c.

A Square-Number is that which is equally equal, or, which is contained under two equal Numbers.

THus the Square-Number (4) is contained under two equal Numbers, viz. (2) and (2) and the Square-Num­ber (9) is contained under two equal Numbers, namely (3) and (3) and so on as in the following Table.

A Table of Squares with their Genetive equal Number.

 Equal Mumber Square 1 into 1 1 2 into 2 4 3 into 3 9 4 into 4 16 5 into 5 25 6 into 6 36 7 into 7 49 8 into 8 64 9 into 9 81 10 into 10 100

Thus the Square-Number (625) is contained under two equal Numbers, viz. (25) and (25) That is to say, [Page 38]

 One of the equal Numbers 25 being multiplyed by the other equal Number. 25 Makes the Product a Square Number, viz. 625

## Eucl. l. 2. Theor. 4. Prop. 4. [...], &c.

If a right Line be cut any wise into two parts, the Square made of the whole Line, is equal both to the Squares made of the Segments, and to twice a Rectangle made of the Parts.

THis holds good likewise in Numbers. For example, Let (25) be the Number of a Right Line, and that di­vided into two parts: viz. (20) the greater, and (5) the lesser. I say,

The Square of the whole (25) is equal to the two Squares of (20) and (5) more by the two Oblongs made of (20) multiplied by (5) as in the opposite Fi­gure may be plainly seen.

 For let (20) be called A And (5) be called B A q: (or 20 multiplyed by 20) makes 400 B q: (or 5 multiplied by 5) makes 25 A, Multiplied by B, that is (20) multiplied by (5) makes 100 A, Multiplied by B, that is (20) multiplied by (5) makes 100 That is to say, these are the parts, which being united are equal to the whole Square 625

These things being premised, I say, that whereas the vulgar Rule directs (after the pointing of the Num­ber 625 whose Square Root is to be extracted) to find out the Square-Root of the last pointed Figure on the left hand, that is (6) and not finding (6) a true Square Number, to take the next to it, viz. (4) whose Square-Root is (2.) I say, the meaning is this; That [Page 40] (6) is in effect (600) which is not a true Square-Number, and the nearest to it is (400) whose Square-Root is (20.)

Again, whereas the vulgar Rule directs to subtract the Square-Number (4) out of (6) and so to find out how many times the double of the first Root is con­tained in the Remainder and the first Figure of the next Square, viz. (22) that is 5 times, But with this provision that there may remain a Number equal to the Square of that (5) as in the example.

[...]

The meaning is this, Having Subtracted A q, or the Square-Number (400) out of the Number (625) there remains (225) which is (A=20) into (B=5) or (100) for one of the Oblongs, and (A=20) into (B=5) for the other Oblong, and B q or (25) for the lesser Square. All which are the very parts of a Square, expressed in the foregoing Proposi­tion of Euclid. Namely.

 (A q) The Square of the greater Segment which is equal to— 400 (B q) The Square of the lesser Segment which is equal to— 25 (A) Into (B) one of the Rectangles—made of the Segments, and equal to 100 (A) Into (B) the other Rectangle made of the Segments and equal to— 100 [Page 41]The Square made of the whole (A▪ [...]) [...] or (25) multiplied by (25) is [...] [...]

And be the Square-Number never so great, both the manner of the Operation, and the Reason of the Ana­lysis, or extraction of its Square-Root▪ is the very same.

But for the better understanding of all that haha been said, let the Practitioner consider w [...]ll the Figure Where he may evidently see, how the Root of every lesser Square being doubled, and an Unite added to it, makes up the next greater Square

Thus twice (3) or the double Root of the next less Square, more by (1) being added to that Square (9) makes up the next greater Square, viz. (16) And twice (5) or the double Root of the next less Square (25) more by (1) that is to say (11) being added to that (25) makes it (36) which is the next greater Square.

# CHAP XVII. The way and Reason of extracting the Square-Root of FRACTIONS.

HE who would rightly comprehend the Nature and Reason of extracting the Square Root of any Fraction, let him consider the Numerator to be a lesser Square-Number, and the Denominator to be a greater Square-Number, and so the lesser to be included in the greater; as for example, if it were demanded to ex­tract the Square-Root of 9/25 the Numerator (9) is to be considered as a lesser Square-Number, and the De­nominator (25) as a greater Square-Number, including or comprehending that lesser or (9 And then all that is to do, is onely to extract the Square-Root of the Nu­merator (9 for a new Numerator, viz. (3) and then the Square-Root of the Denominator (25) for a new De­nominator, [Page 42] viz. (5) As if they were Plain Numbers; That is to say, the Square-Root of 9/25 is ⅗

But now, if neither the Numerator or Denominator be Pure-Square-Numbers, the Operation is somewhat more intricate, though the Reason of it be still the same. As for example, if it were demanded to extract the Square-Root of ⅗ still the Numerator (3) is to be considered as a lesser Square, included in a greater, viz. (5)

That is to say, The Square Root of (3) is [...]

And the Square-Root of (5) is [...]

First, I bring the two Fractions ⅔ and ⅗ to two Fra­ctions of the same Denomination, by the Rule laid down Chap. 8. Pag. 34. and they are 10/15 and 9/15. So then

The Square-Root of (3) is 1 10/15 or 25/15

And the Square-Root of (5) is 2 9/15 or 39/15

# CHAP. XVIII. The Vulgar method of Extracting the Cube-Root.

FOr example, let 15625 be the Number whose Root is to be extracted.

First, it is to be pointed, beginning with the last Fi­gure on the Right hand, and from thence to the 4th from the right hand inclusive, and so to the end, pointing one and leaving two. Thus 15̇625̇ Where Note that the Number of pricks, or points signify the places of the Root sought.

Secondly, the Cube-Root is to be enquired of the last prickt Number on the left hand, (be they three, two, or one) As here the Cube-Root of 15 is to be demand­ed, and being found by the Table of Cube-Numbers Pag. 45. to be 2. I set it in the Quotient for the first [Page 43] Number of the Cube-Root, and then setting the Cube of that (2) viz. (8) under (15) and subtracting the first out of the last, there remains (7) And this is the first step, and the work stands thus.

[...]

In the next place having set (6) the triple of (2 the first found Figure of the Root) under the last Figure save one of the next prickt Number, I square the same (2) and then it is (4) and then I triple that Square (4) and make it (12) and then I set that (12) under the Remainder 762 just as in Division, and this done I ask how many times, (12) is found in the Figures 76 Iust above them, and because it may be found 5 times I set (5) at all adventures for the second Figure of the Root, and then the work stands thus.

[...]

This done, I multiply (the Triple of the Root mul­tiplied by the first Figure of the Root) by this second Figure; That is, I multiply (12) by (5) and set it just under (76).

Again, I multiply the Square of the last Figure, by the Triple of the first, that is (25) by (6) and set it underneath likewise.

In the next place, I Cube that last Figure, (that is I Cube 5) and set that Cube, viz. (125) under the second Cube-number, that is, setting the Unite place of the Cube (125) under the second prickt [Page 44] Figure. And add together those Products, and then the work stands thus.

[...]

Lastly, I compare this (7625) with the Remainder that was left after the deduction of the first Cube-num­ber, 8 and finding them just equal, I turn it off and there remains nothing, the work is at an end, and (5) is the second and the last Figure of the Cube Root.

But now, if the aforesaid Numbers 6000, 1500, & 150, had been greater Numbers, and consequently, being added together, had made a Number greater then 7625) then I must have taken a lesser Number, and consequent­ly the second Figure of the Cube-Root had been not 5) but 4).

Again, if the Numbers 6000, 1500 and 150 being added together, had not made a Number first equal to (7625) That is to say, suppose the Cube-Number given had been 15645, which is more by (20) consequent­ly after the work ended, there had remained 20) which 20 had been the Numerator of a Fraction, whose Denominator is the difference of the Cube of 25 and (26 and that Fraction had been set down thus, 20/1951

And then the answer had been thus, the Cube-Root of (15645) is 25 20/1591

# CHAP. XIX. The Reason and Demonstration of Extract­ing the Cube-Root.

THough the practick of this Extraction may at first sight seem something difficult, yet the Reason and Demonstration of it, will I doubt not, make re­compence.

And in order thereunto I shall first desire the Practi­tioner to consider the Definition of a Cube, as also the Sections of a Cube into its Analytical parts, according to the Rule of Perspective.

## Eucl. l. 7. Defin. XIX [...], [...], &c.

A Cube is that Number which is equally equal, or, which is contained under three equal Numbers.

THus (8) is a Cube which is contained under (2) and (2) and (2).

 Three equal Numbers Cube, 1 into 1 into 1 1 2 into 2 into 2 8 3 into 3 into 1 27 4 into 5 into 4 64 5 into 5 into 5 125 6 into 6 into 6 216 7 into 7 into 7 347 8 into 8 into 8 512 9 into 9 into 9 729

[Page 47] Thus (3) and (3) and (3) make the Cube (27) for (3) multiplied by (3) makes the Square (9) and the Square (9) multiplied again by (3) makes the Cube (27) as is more clear by the foregoing Table.

And so doe (5) and (5) and (5) make the Cube (125) That is, (5) into (5) makes the Square (25) and the Square (25) into (5) makes the Cube (125) &c.

Wherefore to Extract a Cube-Root, is nothing else then to find out a number, which being first multipli­ed into it self, and then into the Product, produces the given Cube-Number. Thus to extract the Cube-Root of (15̇625̇) is to find out the number (25) which be­ing first multiplied into its self (makes 625) and then multiplied into that (625) makes the given Number (15̇625̇)

Now because this construction of the Cube from a sin­gle Root, contributes nothing towards the finding out that Root from a given Cube-Number, therefore was found out by the Antients, that admirable Art of cutting or dividing the Root into two parts, which they therefore called a Binomial Root; and from those two parts they erected 8 solid numbers, whereof the great­est and the least are always two pure Cube numbers, of those two distinct parts, and of the other six Paral­lelepipedons, by which I mean solid Numbers made by multiplying the Square of one Number into another Number, in imitation of the Geometrical Parallelepi­pedons defined by Euclid. lib. 11. Defin. 30. to be a solid Figure contained under six Equilateral Figures, whereof those which are opposite are Parallel.

The three greatest Parallelepipedons are equal one to another, and each of them made by multiplying the Square of the greatest part of the Binomial-Root into the lesser part.

And the three lesser Parallelepipedons are equal to one another, and each of them made, by multiplying the Square of the lesser of the Binomial Root into the great­er part.

Thus in the opposite Figure the whole Root is (5) and divided or cut into a greater part A= 3 and a lesser part B=2

 The Cube of the greater A=3 is equal to 27 The Cube of the lesser B=2 is equal to 8 One of the greater Parallelepipedons, or (Aq into B) is equal to 18 One of the lesser Parallelepipedons or (A into Bq) is equal to 12

 Then To those two Cubes, namely The greater or Ac 27 The lesser or Bc 8 Adding Three of the greater Paralle­lepipedons, viz. (Aq) into (B) 18 (Aq) into (B) 18 (Aq) into (B) 18 And Three of the lesser Parallele­pipedons, viz. (A) into (Bq) 12 (A) into (Bq) 12 (A) into (Bq) 12 The total summe is the entire Cube Num­ber of the Binomial-Root A=3 more by B=2 that is to say (5) and amounteth to 125

Now the Practitioner is to conceive the Unites of the Cube-Number (125) to be as so many Dice, or Cu­bical Unites, and 27 of these being piled one upon another orderly and equally to make up (Ac) and 8 of them to make (Bc) and 18 of them to make one of the greater Parallelepipedons (Aq into B) and 12 of them to make one of the lesser Parallelepipedons (A into Bq) and then all these eight solid Numbers being orderly put together, to make up the entire Cube or (125) And this is the Genesis of a Cube. vid. Fig.

After the very same manner, let the Root (25) be made Binomial and cut into two parts, viz. (20) and (5) and the greater called A, and the lesser B

Then

 The Cube of the greater A=20 is equal to 8000 The Cube of the lesser B=5 is equal to 125 The greater Parallelepipedon, or (Aq into B) is e­qual to 2000 The lesser Parallelepipedon, or (A into Bq) is equal to 500 [Page 49]Then to these two Cubes, namely The greater or (Ac) 8000 The lesser, or (Bc) 125 Adding Three of the greater Parallelepi­pedons, viz. (Aq) into (B) 2000 (Aq) into (B) 2000 (Aq) into (B) 2000 And Three of the lesser Parallelepipe­dons, viz. (A) into (Bq) 500 (A) into (Bq) 500 (A) into (Bq) 500 Summe 15625 The total Summe of all the eight Solids, is the Summe of the entire Cube which amounteth to 15625

All which is consonant to that Theorem of Ramus, (which is in imitation of that of Euclid concerning a Square Number.

## P. RAMI Geometr. Lib. XXIIII, De Cubo pag. 135.

IF a right Line be cut into two Segments, the Cube of the whole shall be equal to the Cubes of the Se­gments, and thrice the double Solid made of the Square of one Segment.

Thus in the last Example, the Square of the greater Segement, multiplied by the lesser, and the Square of the lesser Segment multiplied by the greater Seg­ment, (for that is meant by the double Solid made of the Square of the one Segment into the other Segment) is (2500)

And,

 The triple of that (2500) is 7500 To which adding the two Cubes, or 8125 The total is the entire Cube, viz. 15625

After all this, it will be easie for the meanest ca­pacity to conceive, that while by the vulgar Rules of Extracting the Cube-Root of 15625, the first Cube of (8) is subtracted out of (5) it is indeed (8000 or Ac [...] that is the Cube of the greater Seg­ment subtracted out of (15625) whose Cube-Root is [Page 48] (2) or in appearance, but in reality is (20) or the greater Segment (A)

Again, whereas in the next place the vulgar Rule di­rects, to set down the triple of the first found Figure (2) under the last Figure save one of the next Cube-Number, viz. (2) and then to square that (2) and so make it (4 and then to triple that (4) and so make it (12). And then to find how many times that (12) can be found in the remainder, or (7625) which suppose to be 5 (times) with this provision, that after (12) has been deducted thus 5 times out of (7625) there still remain a Number, either greater or equal to the Square of that (5) that is (25) multiplied by the triple of the first-found Root (2) more by the Cube of 5) that is (125) and if so, the work stands good, and the Cube Root is (25).

The true meaning of all is this.
• 1. The first found Root (2) is Really (20) and is the greater Segment (A)
• 2. The Square of that (20) is (400) and is (A) Square.
• 3. The Triple of that (400) is 1200) and is (A) Square thrice.
• 4. This (1200) is found in (7625) 5 times, which (5) is the lesser Segment (B)
• 5. This (1200) multiplied by (5) or (B) makes (6000) and is thrice (A) Square into (B)
• 6. The Square of that 5) is (25 or (B) Square.
• 7. This (25) or (B) Square multipled by (6) or the triple of (A) makes (1500) and is (A) into (B Square thrice.
• 8. The Cube of this (5) is (125) or (B) Cube. All these being added together, makes up the en­tire Cube.

That is to say,

A, Cube, or the Cube of the greater Segment8000
2. A Square into B, or the triple of the solid Number made of the Square of the great­er Segment multiplied by the lesser Seg­ment6000
[Page 49]3. A, into B Square, or the triple of the solid Number made of the Square of the lesser Segment multiplied by the greater Segment1500
B Cube, or the Cube of the lesser Segment125
The total sum is the intire Cube15625
The given Cube
 Num. 15 625 AB the Binomial 25 Root, viz. A=20 8 — (A) Cube. 7 625 Remainder 60 (A) thrice 1 200 (A) Square thrice
Aggregate of the three for­mer sums and equal to the Remainder a­bove said.
 6 000 (A) Square into (B) thrice, or the triple of the greater Parallel. 1 400 (A) into (B) Square thrice, or the triple of the lesser Parallel. 7 125 (B) Cube. 625

# CHAP. XX. A plain and easie Method of extracting the Square-Root of any Number, (how great soever) without the help of either Mul­tiplication, or Division.

For Example. Let the Square number be—

 2 6 1 6 6 84 34 56 4 2 84 2 76 8 34 5 21 3 13 56 3 13 56 0 00 00

THere are therefore 4 Figures in the Root.

The first, viz. (2) is found by inspection.

The second is had by the Tariffa (A) thus,

Take the double of the first Fi­gure (2) viz. 4, and make a Mul­tiplication Table of it, mingling with the respective Products, the Squares of the 9 Digits as in A. So shall you by inspection find that 276 being the nearest Number to 284, stands over against 6 in the Margint, and therefore (6) is the second Figure of the Root.

 1 4 1 2 8 4 3 1 2 9 4 1 6 1 6 5 2 0 2 5 6 2 4 3 6 7 2 8 4 9 8 3 2 6 4 9 3 6 8 1

 1 5 2 1 2 1 0 4 4 3 1 5 6 9 4 2 0 8 1 6 2 6 0 2 5

 1 5 2 2 1 2 1 0 4 4 4 3 1 5 6 6 9 4 2 0 8 8 6 1 5 2 6 1 0 2 5 6 3 1 3 2 3 6 7 3 6 5 4 9

In the next place, I offer to make the Tariffa B, for the double of the two first Figures 2 and 6, viz 52. But by inspection, I find that it is need­less to fill up the spaces, the very second Product being too great for the Number 834. Therefore (1) must be the next Figure of the Root.

Lastly, I make a Tariffa for the double of the first 3 Figures of the Root, 2, 6, and 1, viz. 522. And by that time I come to the seventh place, I find that 6 is the last Figure, for it gives me 31356, which is the very Number I sought for.

This Method is the most certain, and has the least of difficulty in it of any Methods I ever yet saw.

# CHAP. XXI. Of Proportions Arithmetical, Geometrical, and Musical.

1. Arithmetical Proportion or Habitude, is an Equality of Differences, That is to say, when several Numbers have one and the same Diffe­rence: And this Habitude is two fold.

1. COntinued: When of several Numbers, the second exceeds the first by the very same Unity or Number of Unities, as the third doth the second, and as the fourth doth the third; and so in Infinitum, Thus, 1, 2, 3, 4, 5, 6, 7, 8, &c. differ by an Unit. And 1, 3, 5, 7, 9, &c. have their equal Difference (2.) The orderly proceeding of which Numbers from the lesser to the greater in a Scalal-way, is that which is properly called Arithmetical Progression.

2. Disjunct, when the second exceeds the first by the same Number of Units, as the fourth doth the third, but not as the third doth the second. As for Example, 1, 3, 7, 9, are four Disjunct Arith­metical Proportionals. For (3) exceeds (1) by the same Number of Units as 9) doth (7), but not as (7) doth (3). And thus, 2, 7, 10, 15, are four Proportionals of the same kind, for (7) ex­ceeds (2) by 5, Units, and so doth (15) exceed (10).

[Page 53] 2. Geometrical Proportion or Habitude, is the Equality of Ratio's, that is to say, it is that which shews what part or parts one Number is of another.

Thus, (1, 2, 4, 8) (2, 4, 8, 16,) (3, 6, 12, 24) (4, 8, 16, 32, 64) are Geometrical Proportionals: For in the first Example, as (1) is the half of (2), so is (2) the half of (4), and again, (4) the half of (8), as may be seen in the following Tables.

3. Musical Proportion or Habitude, is when the first Number hath the same Proportion to the third, which the Difference between the first and the second hath to the Difference between the second and the third.

As in (3, 4, 6), (3) is the half of (6), and so is (1) or the Difference between (3) and (4) the half of (2), or the Difference between (4) and (6); and so in (6, 8, 12). But of this in its proper place.

That which is at present to be handled, is the Nature, Properties, and Similitude of the two first kinds of Proportion, Namely Arithmetical and Geometrical, which may be viewed in their several Progressions, by the following Tables.

 1 2 3 4 5 6 7 8 9 10 11 12 2 4 6 8 10 12 14 16 18 20 22 24 3 6 9 12 15 18 21 24 27 30 33 36 4 8 12 16 20 24 28 32 36 40 44 48 5 10 15 20 25 30 35 40 45 50 55 60 6 12 18 24 30 36 42 48 54 60 66 72 7 14 21 28 35 42 49 56 63 70 77 84 8 16 24 32 40 48 56 64 72 80 88 96 9 18 27 36 45 54 63 72 81 90 99 108 10 20 30 40 50 60 70 80 90 100 110 120 11 22 33 44 55 66 77 88 99 110 121 132 12 24 36 48 60 72 84 96 108 120 132 144 13 26 39 52 65 78 91 104 117 130 143 156 14 28 42 56 70 84 98 112 126 140 154 168 15 30 45 60 75 90 105 120 135 150 165 180 16 32 48 64 80 96 112 128 144 160 176 192 17 34 51 68 85 102 119 136 153 170 187 204 18 36 54 72 90 108 126 144 162 180 198 216 19 38 57 76 95 114 133 152 171 190 209 228 20 40 60 80 100 120 140 160 180 200 220 240 21 42 63 84 105 126 147 168 189 210 231 252 22 44 66 88 110 132 154 176 198 220 242 264 23 46 69 92 115 138 161 184 207 230 253 276 24 48 72 96 120 144 168 192 216 240 264 288 25 50 75 100 125 150 175 200 225 250 275 300 26 52 78 104 130 156 182 208 234 260 286 312 27 54 81 108 135 162 189 216 243 270 297 324 28 56 84 112 140 168 196 224 252 280 308 336 29 58 87 116 145 174 203 232 261 290 319 348 30 60 90 120 150 180 210 240 270 300 330 360 31 62 93 124 155 186 217 248 279 310 341 372 32 64 96 128 160 192 224 256 288 320 352 384 33 66 99 132 165 198 231 264 297 330 363 396 34 68 102 136 170 204 238 272 306 340 374 408 35 70 105 140 175 210 245 280 315 350 385 420 36 72 108 144 180 216 252 288 324 360 396 432 37 74 111 148 185 222 259 296 333 370 407 444 38 76 114 152 190 228 266 304 342 380 418 456 39 78 117 156 195 234 273 312 351 390 429 468 40 80 120 160 200 240 280 320 360 400 440 480
 1 1 1 2 3 4 4 9 16 8 27 64 16 81 256 32 243 1 024 64 729 4 096 128 2 187 16 384 256 6 561 65 536 512 19 683 262 144 1 1 1 5 6 7 25 36 49 125 216 343 625 1 296 2 401 3 125 7 756 16 807 15 625 46 536 117 649 68 125 279 216 823 543 340 625 1 675 296 5 764 801 1 703 125 10 051 [...]76 40 353 607 1 1 1 8 9 10 64 81 100 512 729 1 000 4 096 6 561 10 000 32 768 59 049 100 000 262 144 531 441 1 000 000 2 097 152 4 782 969 10 000 000 16 777 216 43 046 721 100 000 000 134 217 728 387 420 489 1 000 000 000 1 1 1 11 12 13 121 144 169 1 331 1 728 2 197 14 641 20 736 28 561 161 051 248 832 371 293 1 771 561 2 985 984 4 826 809 19 487 171 35 831 808 62 748 517 214 358 881 429 981 696 815 730 721 2 357 947 691 5 159 780 352 10 604 499 373
 Geometrical Progression. Arithmetical Progression. 1 1 3 3 9 5 27 7 81 9 243 11 729 13 2 187 15 6 561 17 19 683 19 59 049 21 177 147 23 531 441 25 1 594 323 25 4 782 969 29 14 348 907 31 43 046 721 33 129 140 163 35 387 420 489 37 1 162 261 467 39 3 486 784 401 41 10 460 353 203 43 31 381 059 609 45 94 143 178 827 47 282 429 536 481 49 847 288 609 443 51 2 541 365 828 329 53 7 625 597 484 987 55 22 876 792 454 961 57 68 630 377 364 883 59 205 891 132 094 649 61 617 673 396 283 947 63 1 853 020 188 851 841 65 5 559 060 566 555 523 67 16 677 181 699 666 569 69 50 031 545 098 999 707 71

 Geometrical Progression. Arithmetical Progression. 1 1 2 2 4 3 8 4 16 5 32 6 64 7 128 8 256 9 512 10 1 024 11 2 048 12 4 096 13 8 192 14 16 384 15 32 768 16 65 536 17 131 072 18 262 144 19 524 288 20 1 048 576 21 2 097 152 22 4 194 304 23 8 388 608 24 16 777 216 25 33 554 432 26 67 108 864 27 134 217 728 28 268 435 456 29
 Geometrical Progression. Arithmetical Progression. 10 1 100 2 1 000 3 10 000 4 100 000 5 1 000 000 6 10 000 000 7 100 000 000 8 1 000 000 000 9 10 000 000 000 10 100 000 000 000 11 1 000 000 000 000 12 10 000 000 000 000 13 100 000 000 000 000 14 1 000 000 000 000 000 15 10 000 000 000 000 000 16 100 000 000 000 000 000 17 1 000 000 000 000 000 000 18 10 000 000 000 000 000 000 19 100 000 000 000 000 000 000 20 1 000 000 000 000 000 000 000 21 10 000 000 000 000 000 000 000 22 100 000 000 000 000 000 000 000 23 1 000 000 000 000 000 000 000 000 24 10 000 000 000 000 000 000 000 000 25

In this Progression it is more visible then in any other, how Addition and Substraction in Arithmeti­cal Progression answers to Multiplication and Divi­sion in Geometrical Progression. For, as in Geome­trical Progression, 1000 Multiplied by 100, 000 produce 100, 000, 000; So in Arithmetical Progres­sion, the Number answering to 1000, and 100, 000. viz. (3) and (5) being added together make (8), which answers to the Product 100, 000, 000. Again, as in Geometrical Progression (100, 000 000) being divided by (100, 000) the Quotient is (1000) So in Arithmetical Progression, it from the greatest of their Corespondent Numbers, viz. (8) you Sub­stract any one of the other, viz. (5.) the remainder is (3) and answereth to (1000.)

## I. Reflection. [...]. Prop. Tertia.

β.α..γ..δ..ε. B.A..G..D..E.

[...], &c. If Numbers (how many soever they be) exceed one another by an equal Interval, then the Interval between the great­est, and the least, is Multiplex of that equal Interval, according to the Multitude of the Numbers propound­ed, less by one.

Let the Numbers propounded be four, viz. |BA1| |BG3| |BD5| |BE7| whose common In­terval is equal to |AG2|

By the Hypothesis |AE6| is the Interval between the greatest |BE7| and the least |BA1|: And likewise the three Numbers |AG2| |GD2| |DE2| and every of them equal to the Common Difference, and equal the one to the other. And the multitude of them equal to the Multitude of the Numbers given less by one (viz. the least.) And lastly, the Aggregate of these three Numbers is equal to the Interval |AE6| (the parts united being equal to the whole.)

That is to say, |AE6| or the Interval between the greatest |BE7| and the least |BA1| is Multiplex of the common Interval

## I. Reflection

IF Numbers (how many soever they be) contain the one the other by an equal Ratio; Then the greatest of those Numbers is Multiplex of the Powers of the Denomination of that equal Ratio Multiplyed by the least, according to the Multitude of the Num­bers given, less by one

This Reflection I have framed for Geometrical Proportionals, in imitation of that Diophantus for Arithmetical Proportionals.

Let the Numbers given be Four, viz. 2, 6, 18, 54.

And let the Denominator of the Ratio be 3.

Then by the Hypothesis, the first Multiplyed by (3) is equal to the second; and the second Multi­plyed by (3) is equal to the third; and the third Multiplyed by (3) is equal to the fourth: And so in Infinitum.

That is to say,

• 1. The first Term (2) is equal to 2
• 2. The 2d Term (6) is equal to 2 into 3 1st power of the Ratio, or the single Ratio.
• 3. The 3d Term (18) is equal to 2 into 3 into 3 2d power of the Ratio, or the Ratio Squared.
• 4. The 4th Term (54) is equal to 2 into 3 into 3 into 3 3d power of the Ratio, or the Ratio Cubed.

That is to say, in a Symbolical way.

Let there be any Number of Proportional Num­bers, as, A, B, C, D, E, F, &c. And the Denomi­nation of the Ratio be R.

[Page 60] |AG2| according to the Multitude of the Numbers propounded less by one, Which was to be Demon­strated.

In a Symbolical way, Thus,

Then let the Terms be A, B, C, D, and the Common Difference E.

Thus, [...]

Or thus, [...]

In plain English thus,
• 1st Term (1) is equal to 1
• 2d Term (3) is equal to 1 more by 2.
• 3d Term (5) is equal to 1 more by 2 more by 2.
• 4th Term (7) is equal to 1 more by 2 more by 2 more by 2.

That is to say, by the Hypothesis, the greatest Term is equal to the least, and as many Differences as there are more Terms besides the least; there­fore the greatest Term less by the least is Multiplex of the Difference according to the Number of Terms less by one.

As was to be Demonstrated.

• [Page 61]A=A
• B=A into R
• C=A into R into R
• D=A into R into R into R
• E=A into R into R into R into R
• F=A into R into R into R into R into R
• G=A into R into R into R into R into R into R

Or thus,

 Powers of R. A=A B=A into R 1 C=A into R Square 2 D=A into R Cube 3 E=A into R Squared Square 4 F=A into R Squared Cube 5 G=A into R Cubick Cube 6

As this Reflection carryes its evidence along with it, so is it of admirable Speculation and Use, as hereafter will appear.

## II. Reflection.

WHere three Numbers are three Arithmetical Proportionals, the Sum of the two Extreams is equal to the Double of the Mean.

Let the three Numbers be (2, 4, 6)

And the Common Difference be (2)

Then by the First Reflection.

The First Term is 2

The second Term is 2 more by the Difference once.

The third Term is 2 more by the Difference twice.

Whence it is evident and obvious, That twice (2) more by twice the Difference, is both the sum of the first and third Terms, and also the double of the second or Mean.

Which was to be Demonstrated.

## II. Reflection. Eucl. Lib. 7. Prop. 20.

[...], &c. If there be three Numbers in proportion, the Number contained under the Extreams, is equal to the Square made of the Mean; And if the Number contained under the Extreams be equal to the Square of the Mean, those three Numbers shall be in Proportion.

Let the three Numbers be (2, 6, 18)

And the Equal Ratio (3.)

Then by the first Reflection.

The first Term is 2 A

The second Term is 2 into (3) A × R

The third Term is 2 into (the Square of 3) A × R × R

Whence it is evident and obvious in the first place, that the first Term drawn into the third is equal to 2 into 2, into the Square of 3.

Again it is evident, That the Square of the second Term is equal to the Square of 2 into the Square of 3.

But these two are equal; That is to say, (2) into (2) into (the Square of 3) = 36 (the Square of 2) into (the Square of 3) = 36 Ergo,

The Product of the First Multiplyed by the Third, is equal to the Square of the Second or Mean. Which was to be demonstrated.

## III. Reflection.

WHere four Numbers are four Arithmetical Proportionals, the Sum of the first and fourth, is equal to the Sum of the second and third.

First, let them be four continued Proportionals, viz. (4, 12, 20, 28)

And their Common Difference (8)

Then by the second Reflection.

• The first Term is 4
• The 2d Term is 4 more by once the Difference.
• The 3d Term is 4 more by twice the Difference.
• The 4th Term is 4 more by thrice the Difference.

Secondly, let them be four discontinued Propor­tionals, viz. (4, 12, 30, 38.)

And their Common Difference (8)

Then by the second Reflection, and the Hy­pothesis.

• The first Term is 4
• The 2d Term is 4 more by once the Difference.
• The 3d Term is 30
• The 4th Term is 30 more by once the Difference.

## III. Reflection. Eucl. Lib. 7. Prop. 19.

[...], &c. If there be four Numbers in proportion, the Number produced of the first and fourth, is Equal to the Number which is pro­duced of the second and third. And if the Number produced of the first and fourth be Equal to that pro­duced of the second and third: Those four Numbers shall be in proportion.

First let them be four continued Proportionals. viz. (2, 6, 18, 54) and the Equal Ratio (3).

Then by the Second Reflection.
• The first Term is 2.
• The second Term is 2 into (3).
• The third Term is 2 into the Square of (3).
• The fourth Term is 2 into the Cube of (3).

Secondly, let them be four Disjoynt Proportionals. viz. (2, 6, 54, 162) and the Equal Ratio (3).

Then by the second Reflection, and the Hypothesis.
• The first Term is 2.
• The second Term is 2 into (3).
• The third Term is 54.
• The fourth Term is 54 into (3).

[Page 66] In the first Example it is evident,

That twice (4) more by thrice the Difference (24), is both the Sum of the Extreams, and also the Sum of the second and third, viz. 32.

In the second Example it is as evident,

That (4) more by (30) more by once the Dif­ference, is both the Sum of the Extreams, and also of the second and third Terms, viz. 42.

[Page 67] In the first Example it is evident by the Multi­plication of Powers.

That the Square of (2) Multiplyed by the Cube of (3) is both the Product of the Extreams, and also of the two Means.

In the second Example it is as evident,

That (2 into 3) Multiplyed by (54) is both the Product of the Extreams, and also of the Means.

## IV. Reflection.

IN all Continued Arithmetical Progressions, (how many soever the Terms be) the Sum of the Ex­treams is equal to the Sum of any two of the other Terms equidistant from the Extreams, and to the double of the Middle Term, in case the Number of Terms be odd.

Let the Number of Terms be Seven.

viz. 3, 6, 9, 12, 15, 18, 21, and the Common Difference (3).

First by the Hypothesis.

The four Numbers (3, 6, 18, 21) are four Pro­portionals.

Therefore by the fourth Reflection.

Their two sums, viz

• 3 more by 21
• 6 more by 18

are Equal.

Secondly, by the same Hypothesis.

The four Numbers (6, 9, 15, 18,) are four Proportionals.

Therefore by the fourth Reflection.

Their two sums, viz.

• 6 more by 18
• 9 more by 15

are Equal.

Lastly, by the same Hypothesis.

The three Numbers (9, 12, 15) are three Proportionals.

## IV. Reflection.

IN all Continued Geometrical Progressions (how many soever the Terms he) the Product of the Extreams is equal to the Product of any two of the other Terms equidistant from the Extreams, and to the Square of the Middle Term, in case the Number of Terms be odd.

Let the Number of Terms be Seven.

Viz. (2, 6, 18, 54, 162, 486, 1458); and the equal Ratio (3).

First, by the Hypothesis.

The four Numbers (2, 6, 486, 1458) are four Proportionals.

Therefore by the fourth Reflection.

Their two Pro­ducts, viz.

• 2 into 1458
• 6 into 486

are Equal.

Secondly, by the same Hypothesis.

The four Numbers (6, 18, 162, 486) are four Proportionals.

Therefore by the fourth Reflection.

Their two Pro­ducts, viz.

• 6 into 486
• 18 into 162

are Equal.

Lastly, by the same Hypothesis:

The Three Numbers (18, 54, 162) are three Proportionals.

[Page 70] Therefore by the third Reflection.

• 9 more by 15
• 12 doubled

are Equal.

Therefore by the 1st Ax. Euclid.

• 3 more by 21
• 6 more by 18
• 9 more by 15
• 12 doubled

are all Equal one to a­nother. As was to be Demonstrated.

[Page 71] Therefore by the third Reflection. 18 into 162 and 54 Squared, are Equal.

Wherefore,

• 2 into 1458
• 6 into 486
• 18 into 162
• 54 Squared

do make 2916, and con­sequently are all Equal one to another: Which was to be Demonstrated.

## The foregoing Reflections applyed to the Golden Rule, or Rule of Three.

The Rule of Three, or the Golden Rule.

THis Rule is either Single, or Compound;

1. The Single Rule of Three.

The Single Rule of Three, is when three Num­bers are Given, and a fourth Proportional De­manded.

Now this Single Rule, is either Direct, or Inverse.

1. Direct.

The Single Rule of Three Direct, is when three Numbers are Given, and a fourth is Demanded, which bears the same proportion to the Third, as the Second bears to the First.

Example.

If 4 Acres of Ground cost 80 Pound, what will 8 Acres of the same Ground cost?

To understand this Operation, look back to the third Reflection upon Geometrical Proportion, and you shall find it Demonstrated, that if there be four Proportionals, the Product of the First and Fourth is equal to the Product of the Second and Third.

Wherefore in this Example.

If the Product of the Second and Third Term, [Page 73] viz. (640) be Divided by the First, namely (4) The Quotient, viz. (160) is the fourth Proportional sought.

And for a Proof of this,

Let the fourth found Term (160) being Mul­tiplyed by the first (4); the Product will be the same with the Product of the third Multiplyed by the second, viz. 640.

The Argumentation is plainly thus,

If the Product of (80 by 8) be equal to the Product of (4 by the unknown Number.)

Then,

The Quotient will be the very same, whether I Divide the Product of (80 by 8,) or whether I Divide the Product of (4 by the unknown Num­ber) by 4. For either of the Products being 640, the Quotient must needs be 160.

But now it is manifest, that if

I Multiply 4 by 160, and Divide the Product back again by 4, it will give 160 for the Quotient. Because, whatever Multiplication doth, is again undone by Division. And this is the true and genuine Reason of the Operation in this Rule of Three.

[Page 74] That is,

As (4) is found in (80) just 20 times, so is (8) found in (160) just 20 times.

Or thus,

As 4 Multiplyed by 20 makes 80, so 8 Multi­plyed by 20 makes 160.

2. Example.

If 80 l. will buy 4 Acres, what will 160 l. buy?

The Answer will be found as before.

By Dividing the Product of (4 into 160) viz. (640) by (80) For then the Quotient will be 8.

For as 80 contains 4, twenty times, so 160 be­ing Divided by 20, the Quotient is 8.

2. Inverse.

The Single Rule of Three Inverse is when there are three Numbers given, and a fourth demanded, which bears the same Proportion to the second, as the third doth to the first.

Example.

If a quantity of Hey will keep 8 Horses 12 Days, How many Days will the same quantity keep 16 Horses.

[Page 75] Here it is observable, that most of those who have hitherto Treated of this Rule of Three, puzzle both young Learners and themselves, with this distinction of Direct and Inverse, as though they were two distinct and different things, whereas in truth they are the very same thing, only care must be taken how to place them. For the true un­folding of this Question, is this,

Look what proportion 16 Horses bear to 8 Horses, so do 12 Days bear to a fourth Number of Days. And they ought to be placed thus,

And then the operation is the very same as be­fore; for the Product of 8 by 12 is (96) which being Divided by 16, the Quotient is (6), which the fourth Number sought.

2. The Double Rule of Three.

And this is

• 1. Direct.
• 2. Inverse.

1. Direct.

THe Double Rule of Three is when more then three Terms are given; as the ordinary Books of Arithmetick teach us.

1. Example.

If 4 Men spend 19 Pound in three Months; how many Pounds will 8 Men spend in nine Months?

The Resolution of this Question is thus per­formed.

1. If four Men spend 19. what will 8 Men spend.

[Page 77] 2. If 38 l. be spent by any Number of Men in 3 Months, how many Pounds will be spent by such a Number of Men in 9 Months?

2. Example.

If 9 Bushels of Oats serve 8 Horses 12 Days, how many Days will 24 Bushels last 16 Horses.

1. If 9 Bushels last 12 Days, how many Days will 24 Bushels last?

[Page 78] 2. As 16 Horses is to 8 Horses, so is 32 Days to 16 Days.

And this is the whole Intrigue of the Golden Rule, or the Rule of Three. The which being rightly understood, (together with the foregoing Reflections on Arithmetical and Geometrical Pro­portion) those other Rules of Fellowship, and Alligation, as likewise the Rule of False, will not be at all difficult. And therefore I think it needless to multiply Examples.

## The Diameter of any Circle being given in Integers, to find the Peripherie, and the Square-Root of the Area, in infinitum, without the help of either Multiplication, Division, or Extraction of the Root.

FOr Example, Let the given Diameter of a Circle be 351. And let it be required to find the Pe­ripherie (or Circumference.) First I set down (300,) under that (50,) and beneath that 1, as in the fol­lowing Operation. This done, I first seek the Peri­pherie of a Diameter of (3) in the subsequent Table, viz. 9, 42477795, and because it is the place of Hundreds, I add thereto on the right hand two Ci­phers (00.) Then, under that Sum I set the Number of the Peripherie of (5.) adding to it one Cypher, because it is the place of Tens (viz. for 50.) Lastly, I place under the two Sums abovesaid the Peripherie of (1.) without adding any Cypher, because it is the place of Ʋnits. And adding these 3 Sums together, (evermore distinguishing the Integers from the Fractions, after the 8th place from the right hand, with a Comma, or Line of Seperation) the Sum, or Aggregate (viz. 1102, 6990215,) is the true Peri­pherie of that Circle, whose Diameter is 351.

 300—942 4777, 9500 50—157 0796, 3250 1—3 1415, 9265 351 sum [...] 1102 6990, 2015

After the very same manner, if it were required to give the Square-root of the Area of that very Circle, whose Diameter is 351. I add the 3 following Sums, and the Aggregate is what I desire. For Example.

 300—265 8680, 7700 50—44 3113, 4620 1—0 8862, 2692 351 sum 311 0656, 5012

If the first place of the given Diameter be of Thousands, there must be 3 Cyphers; if of Tens of Thousands, there must 4 Cyphers be added on the right hand, and so in Infinitum.

 Diamet. Peripherie. Square-root of the Area 1 3 1415, 9265 8862, 2692 2 6 2831, 8530 1 7724, 5385 3 9 4247, 7795 2 6586, 8077 4 12 5663, 7060 3 5449, 0770 5 15 7079, 6325 4 4311, 3462 6 18 8495, 5590 5 3173, 6155 7 21 9911, 4855 6 2035, 8847 8 25 1327, 4120 7 0898, 1540 9 28 2743, 3385 7 9760, 4232

But in case of Fractions, as for Example, if the Diameter had been given 351 ¼, and the Peripherie were demanded: The way is, to take ¼ of 3, 1415, 9265. (viz. 0, 7853, 9816) and to add it to 1102, 6990, 2015, which makes 1103, 4844, 1831 for the Peripherie of that Circle whose Diameter is is 351 ¼. And the same method is to be used for any Fraction whatsoever.

 Circum. Diameter. Square-root of the Area 1 3183, 0988 2820, 9479 2 6366, 1977 5641, 8958 3 9549, 2965 8462, 8437 4 1 2732, 3954 1 1283, 7916 5 1 5915, 4942 1 4104, 7395 6 1 9098, 5931 1 6925, 6875 7 2 2281, 6920 1 9746, 6354 8 2 5464, 7908 2 2567, 5833 9 2 8643, 8897 2 5388, 5312

This Table is to be used after the very same man­ner with the former.

## A Perpetuall Almanack invented by S. Morland 1650.

 To find the Dom: Letter for ever. C E G B D F A B D F A C E G A C E G B D F GF BA DC FE AG CB ED 1000 1200 1400 1600 1100 1300 1500 1616 20 24 28 32 36 40 44 48 52 56 60 64 68 72 76 80 84 88 92 96 1700 4 8 12 16 20 24 28 32 36 40 44 48 52 2400 1900 2100 2300 1800 2000 2200 To find what day of the Week each Month begins. A B C D E F G Octob: 31 May. 31 August 31 March 31 Iune. 30 Septem 30 Aprill. 30 Ianu: 31 Novem 30 Decem: 31 Iuly: 31 Febru: 2 8/9 To find the day of ye Month 1 8 15 22 29 2 9 16 23 30 3 10 17 24 31 4 11 18 25 5 12 19 26 6 13 20 27 7 14 21 28

## AN EXPLANATION OF THE Perpetual Almanack.

THis Almanack was first intend­ed to be as short and compen­dious as was possible, and to be Graven on a small plate of Silver, about the breadth of a Shilling, and so portable toge­ther with Money: but having now design'd it for a small pocket Book, I have thought it more convenient to divide it into three distinct Tables, which are much more plain and easie, that so the use thereof may be obvious to the meanest capacity. And when the three following Tables and their use are throughly understood, This Almanack will need no explanation, and may be used sometimes as well as the other three Tables.

### The use of the Table in Page 6.

By the Table in Page 6. you may immediately find the Dominical Letter from the first year of our Lord, to the year 3400, &c. so long as the world shall last.

For example, For any year of Our Lord under 100. they are exprest in the Marginal Columnes of Numbers, and are answered in the first Columne of Letters. Thus for the years of Our Lord 28, 56, and 84. The Domi­nical Letters are DC. For the years 1, 29, 57, and 85. the Dominical Letter is B. For the years 11, 39, 67, and 95. the Dominical Letter is D.

[Page 2] Now for all the even Hundreds or Thousands from 100. to 3400. (which are all Leap-years) the Domi­nical Letters are to be found above them in the 1, 2, 3, 4, 5, 6, and 7th Columne.

And for any other Numbers, the Rule is for ever this; Where the Marginal Number (equal to the Fracti­on or part of an Hundred sought) meets with the Columne where that Number of Hundreds or Thousands (which is the other part of the Number sought) is exprest; That is the Dominical Letter (or Letters) for that year.

For example, Let the Dominical Letter be demanded for the year of Our Lord 1672. The number of Hun­dreds is, 1600. and is found in the third Columne. And the broken part (72) among the Marginal Numbers meets with that third Columne in the Letters GF. which are the Dominical Letters for that year, namely the first of them (G.) from the first of January to S. Matthias Eve, the twenty fourth of February, and the other Letter F. to the years end, which is a constant Rule for all Leap-years.

If the Dominical Letter be demanded for 1481. The Hundreds are 1400. and are found in the first Co­lumne, the broken part of an Hundred (81) meets with that Columne in G. which is the Dominical Letter for that year.

So if the Dominical Letter be demanded for 1349, 1300. is found in the 7 Columne, where the broken part (49) among the Marginal Numbers; meets in the Letter D. which is the Dominical Letter for that year.

Again, Let the Dominical Letter be demanded for the year 1729. 1700. is found in the 4 Columne, and 29. meets with that Columne in E.

And so for any year past or to come.

### The use of the Table in page 7.

The use of this Table is to find readily, both upon what day of the week any moneth of the year begins for ever. But also how many days each month con­tains, [Page 3] which is to be sound under the name of each re­spective Moneth.

For example, I desire to know upon what day of the week the moneth of April begins, in the year 1623.

First I find by the Table in page 6. that the Domini­cal Letter for that year is E. Then I guide my Eye down that Columne of the Table in page 7. where E. is at the top, and at the same time observe where April is in the Margent, and where the Line of April meets with the Columne E. there I find (Tuesday) which gives me to understand that April in the year 1623. begins on a Tuesday.

By the same Rule

The moneth of May in the year 1615. begins on a Munday.

The same moneth in the year 1616. began on a Wednesday; where it is to be noted that forasmuch as 1616 is a Leap year, I make use of the 2 Letter (F.) for the Table in page 7. which must always be ob­served in Leap-years.

The moneth of September in the year 1537. began on a Saturday.

The moneth of July in the year 1471, began on a Monday.

The Moneth of December in the year 1642. began on a Thursday.

The moneth of August in the year 1781. will begin on a Sunday.

And so in all the rest.

### The Use of the Table in Page 8.

The use of this Table (having first found upon what day of the week any Moneth begins) is to know readily the day of the Moneth.

Suppose it to be Thursday, and the beginning of October in the year 1672.

By the two former Tables I find, that the Moneth of October 1672. begins on a Tuesday; wherefore I [Page 4] look among the Tables in page 8. till I find a Ta­ble that begins with a Tuesday; and finding that the Thursdays of that Moneth are 3, 10, 17, 24, and 31. and that it is yet but the beginning of the Moneth, I conclude that it must needs be the 3 day of the Moneth.

Where it is to be observed, that if I cannot by some Marks or Circumstances help my memory, to know at least what week of the Moneth it is, no Alma­nack in the World can inform or tell me what day of the Moneth it is.

### Two excellent Uses of the Ta­ble, in Page 8.

1. By it may be readily found what day of the week, any day of the Month was, or will be, of any year past, or to come.

For example; Such a man was born, or such an acti­on was done, or such a Letter was written the fourth of September in the year 1618. and it is demanded what day of the week it was?

1. By the Table in Page 6. I find the Dominical Let­ter for that year to be D.

2. By the Table in Page 7. I find that September in that year began on a Tuesday.

3. And by the Table in page 8. I find that the fourth of September was the first Friday of that Moneth. And so may any other day of any week, of any other Month, of any other year past, or to come, be known.

[Page 5] 2. By it may be readily found what day of the Month, was or will be, of any day, of any week, of any month, of any year past, or to come.

For example; Suppose such a Man was born, or such an Action done, or such a Letter written, upon the first Friday of September, in the year 1618.

Having found by the foregoing directions, that Sep­tember in that year began on a Tuesday, I find that the first Friday of that Moneth, was the fourth day of the same Moneth.

And so may be readily found what day of the Month, was or will be, any day of any week, of any month, of any year, past or to come.

Al which are of excellent and daily use for all Ministers of State, and for all Merchants in their correspondencies.

 1 2 3 4 5 6 7 DC ED FE GF AG BA CB 000 100 200 300 400 500 600 700 800 900 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 2000 2100 2200 2300 2400 2500 2600 2700 2800 2900 3000 3100 3200 3300 3400 0 28 56 84 DC ED FE GF AG BA CB 1 29 57 85 B C D E F G A 2 30 58 86 A B C D E F G 3 31 59 87 G A B C D E F 4 32 60 88 FE GF AG BA CB DC ED 5 33 61 89 D E F G A B C 6 34 62 90 C D E F G A B 7 35 63 91 B C D E F G A 8 36 64 92 AG BA CB DC ED FE GF 9 37 65 93 F G A B C D E 10 38 66 94 E F G A B C D 11 39 67 95 D E F G A B C 12 40 68 96 CB DC ED FE GF AG BA 13 41 69 97 A B C D E F G 14 42 70 98 G A B C D E F 15 43 71 99 F G A B C D E 16 44 72 ED FE GF AG BA CB DC 17 45 73 C D E F G A B 18 46 74 B C D E F G A 19 47 75 A B C D E F G 20 48 76 GF AG BA CB DC ED FE 21 49 77 E F G A B C D 22 50 78 D E F G A B C 23 51 79 C D E F G A B 24 52 80 BA CB DC ED FE GF AG 25 53 81 G A B C D E F 26 54 82 F G A B C D E 27 55 83 E F G A B C D 1 2 3 4 5 6 7
 A B C D E F G Ian 31 Sund: Satur: Fryd: Thur: Wedn: Tuesd: Mun: Feb 28 Wedn: Tuesd: Mund. Sund. Satur. Fryd: Thur: Mar 31 Wedn: Tuesd: Mund: Sund: Satur: Fryd: Thur: Apr 30 Satur: Fryd: Thur: Wedn. Tuesd: Mund: Sund: May 31 Mund: Sund: Satur: Fryd: Thur: Wedn: Tuesd: Iun 30 Thur: Wedn: Tuesd: Mund. Sund: Satur. Fryd: Iuly 31 Satur: Fryd: Thur: Wedn: Tuesd: Mund. Sund: Aug 31 Tuesd: Mund. Sund: Satur. Fryd: Thur: Wedn: Sep 30 Fryd: Thur: Wedn: Tuesd. Mund. Sund: Satur. Oct 31 Sund: Satur. Fryd: Thur: Wedn: Tuesd: Mund. Nou 30 Wedn: Tuesd: Mund: Sund: Satur. Fryd: Thur: Dec 31 Fryd: Thur: Wedn: Tuesd: Mund. Sund. Satur.
The 7 Varieties of the last Table Pag: 1 for finding ye day of the Month.
 Wedn: 1 8 15 22 29 Thursdays 2 9 16 23 30 Frydays 3 10 17 24 31 Saturdays 4 11 18 25 Sundays 5 12 19 26 Mundays 6 13 20 27 Tuesdays 7 14 21 28
 Sund: 1 8 15 22 29 Mundays 2 9 16 23 30 Tuesdays 3 10 17 24 31 Wednesdays 4 11 18 25 Thursdays 5 12 19 26 Frydays 6 13 20 27 Saturdays 7 14 21 28
 Thursd: 1 8 15 22 29 Frydays 2 9 16 23 30 Saturdays 3 10 17 24 31 Sundays 4 11 18 25 Mundays 5 12 19 26 Tuesdays 6 13 20 27 Wednesdays 7 14 21 28
 Mund: 1 8 15 22 29 Tuesdays 2 9 16 23 30 Wednesdays 3 10 17 24 31 Thursdays 4 11 18 25 Frydays 5 12 19 26 Saturdays 6 13 20 27 Sundays 7 14 21 28
 Fryd: 1 8 15 22 29 Saturdays 2 9 16 23 30 Sundays 3 10 17 24 31 Mundays 4 11 18 25 Tuesdays 5 12 19 26 Wednesdays 6 13 20 27 Thursdays 7 14 21 28
 Tuesd: 1 8 15 22 29 Wednesdays 2 9 16 23 30 Thursdays 3 10 17 24 31 Frydays 4 11 18 25 Saturdays 5 12 19 26 Sundays 6 13 20 27 Mundays 7 14 21 28
 Saturd: 1 8 15 22 29 Sundays 2 9 16 23 30 Mundays 3 10 17 24 31 Tuesdays 4 11 18 25 Wednesdays 5 12 19 26 Thursdays 6 13 20 27 Frydays 7 14 21 28

[Page 9] To find out the Prime or Golden Number for ever Divide the year of the Lord by (19) and to the Remaynder after the Division add (1) the sum: is the Prime for that year

Example Thus (1671) divided by (19) leaues (18) to which adding (1) makes it (19) for the Prime of yt year

[...]

 1—19 6—114 2—38 7—133 3—57 8—152 4—76 9—171 5—95
 100 200 300 400 500 600 700 800 900 6 11 16 2 7 12 17 3 8 2000 2100 2200 2300 2400 2500 2600 2700 2800 1 20 39 58 77 96 7 12 17 3 8 13 18 4 9 2 21 40 59 78 97 8 13 18 4 9 14 19 5 10 3 22 41 60 79 98 9 14 19 5 10 15 1 6 11 4 23 42 61 80 99 10 15 1 6 11 16 2 7 12 5 24 43 62 81 11 16 2 7 12 17 3 8 13 6 25 44 63 82 12 17 3 8 13 18 4 9 14 7 26 45 64 83 13 18 4 9 14 19 5 10 15 8 27 46 65 84 14 19 5 10 15 1 6 11 16 9 28 47 66 85 15 1 6 11 16 2 7 12 17 10 29 48 67 86 16 2 7 12 17 3 8 13 18 11 30 49 68 87 17 3 8 13 18 4 9 14 19 12 31 50 69 88 18 4 9 14 19 5 10 15 1 13 32 51 70 89 19 5 10 15 1 6 11 16 2 14 33 52 71 90 1 6 11 16 2 7 12 17 3 15 34 53 72 91 2 7 12 17 3 8 13 18 4 16 35 54 73 92 3 8 13 18 4 9 14 19 5 17 36 55 74 93 4 9 14 19 5 10 15 1 6 18 37 56 75 94 5 10 15 1 6 11 16 2 7 19 38 57 76 95 6 11 16 2 7 12 17 3 8
 1000 1100 1200 1300 1400 1500 1600 1700 1800 1900 13 18 4 9 14 19 5 10 15 1 2900 3000 3100 3200 3300 3400 3500 3600 3700 3800 1 20 39 58 77 96 14 19 5 10 15 1 6 11 16 2 2 21 40 59 78 97 15 1 6 11 16 2 7 12 17 3 3 22 41 60 79 98 16 2 7 12 17 3 8 13 18 4 4 23 42 61 80 99 17 3 8 13 18 4 9 14 19 5 5 24 43 62 81 18 4 9 14 19 5 10 15 1 6 6 25 44 63 82 19 5 10 15 1 6 11 16 2 7 7 26 45 64 83 1 6 11 16 2 7 12 17 3 8 8 27 46 65 84 2 7 12 17 3 8 13 18 4 9 9 28 47 66 85 3 8 13 18 4 9 14 19 5 10 10 29 48 67 86 4 9 14 19 5 10 15 1 6 11 11 30 49 68 87 5 10 15 1 6 11 16 2 7 12 12 31 50 69 88 6 11 16 2 7 12 17 3 8 13 13 32 51 70 89 7 12 17 3 8 13 18 4 9 14 14 33 52 71 90 8 13 18 4 9 14 19 5 10 15 15 34 53 72 91 9 14 19 5 10 15 1 6 11 16 16 35 54 73 92 10 15 1 6 11 16 2 7 12 17 17 36 55 74 93 11 16 2 7 12 17 3 8 13 18 18 37 56 75 94 12 17 3 8 13 18 4 9 14 19 19 38 57 76 95 13 18 4 9 14 19 5 10 15 1
 Domin [...] Letter Golden Number From Christmas to Shrove sund: Shrove sunday Easter day A 2 5 13 16 6 weeks Febr 5 Mar 26 7 10 15 18 7 weeks Febr 12 Apr 2 1 4 9 12 8 weeks Febr 19 April 9 3 6 11 14 17 9 weeks Febr 26 Apr 16 8 19 10 weeks Mar 5 Apr [...]3 B 2 5 13 16 6 weeks 1 day Febr 6 Mar 27 4 7 10 15 18 7 weeks 1 day Febr 13 Apr 3 1 9 12 17 8 weeks 1 day Febr 20 Apr 10 3 6 11 14 9 weeks 1 day Febr 27 Apr 17 8 19 10 weeks 1 day Mar 6 Apr 2 C 2 5 10 13 6 6 weeks 2 days Febr 7 Mar 28 4 7 15 18 7 weeks 2 days Febr 14 April 4 1 6 9 12 7 8 weeks 2 days Febr 21 Apr 11 3 11 14 19 9 weeks 2 days Febr 28 Apr 18 8 10 weeks 2 days Mar 7 Apr 25 D 16 5 weeks 3 days Febr 1 Mar 28 2 5 10 13 6 weeks 3 days Febr 8 Mar 29 4 7 12 15 18 7 weeks 3 days Febr 15 Apr 5 1 6 9 17 8 weeks 3 days Febr 23 Apr 12 3 8 11 14 19 9 weeks 3 days Mar 1 Apr 19 E 5 16 5 weeks 4 days Febr 2 Mar 23 2 10 13 18 6 weeks 4 days Febr 9 Mar 30 1 4 12 15 7 weeks 4 days Febr 16 april 6 6 9 14 17 8 weeks 4 days Febr 23 apr 13 3 8 11 19 9 weeks 4 days Mar 2 apr 20 F 5 16 5 weeks 5 days Febr 3 Mar 24 2 7 10 13 18 6 weeks 5 days Febr 10 Mar 31 1 4 12 15 7 weeks 5 days Febr 18 apr 7 3 6 9 14 17 8 weeks 5 days Febr 24 apr 14 8 11 19 9 weeks 5 days Mar 3 apr 21 G 5 13 16 5 weeks 6 days Febr 4 Mar 25 2 7 10 18 6 weeks 6 days Febr 11 apr 1 1 4 9 12 15 7 weeks 6 days Febr 18 apr 8 3 6 14 17 8 weeks 6 days Febr 25 apr 15 8 11 19 9 weeks 6 days Mar 4 apr 22
 Dom: Letter Golden Number Rogat: Asens: Whit: Trinity Advent Sund: day Sund: Sund: Sund: A 2. 5. 13. 16. Apr. 30 May. 4 May. 14 May. 21 Dec. 3 7. 10. 15. 18. May. 7 May. 11 May. 21 May. 28 Dec. 3 1. 4. 9. 12. May. 14 May. 18 May. 28 Iun. 4 Dec. 3 3. 6. 11. 14. 17 May. 21 May. 25 Iun. 4 Iun. 11 Dec. 3 8. 19. May. 28 Iun. 1 Iun. 11 Iun. 18 Dec. 3 B 2. 5. 13. 16. May. 1 May. 5 May. 15 May. 22 Nov. 27 4. 7. 10. 15. 18. May. 8 May. 12 May. 22 May. 29 Nov. 27 1. 9. 12. 17. May. 15 May. 19 May. 29 Iun. 5 Nov. 27 3. 6. 11. 14. May. 22 May. 26 Iun. 5 Iun. 12 Nov. 27 8. 19. May. 29 Iun. 2 Iun. 12 Iun. 19 Nov. 27 C 2. 5. 10. 13. 16. May. 2 May. 6 May. 16 May. 23 Nov. 28 4. 7. 15. 18. May. 9 May. 13 May. 23 May. 30 Nov. 28 1. 6. 7. 9. 12. May. 16 May. 20 May. 30 Iun. 6 Nov. 28 3. 11. 14. 19. May. 23 May. 27 Iun. 6 Iun. 13 Nov. 28 8. May. 30 Iun. 3 Iun. 13 Iun. 20 Nov. 28 D 16. Apr. 26 Apr. 30 May. 10 May. 17 Nov. 29 2. 5. 10. 13. May. 3 May. 7 May. 17 May. 24 Nov. 29 4. 7. 12. 15. 18. May. 10 May. 14 May. 24 May. 31 Nov. 29 1. 6. 9. 1. 7. May. 17 May. 21 May. 31 Iun. 7 Nov. 29 3. 8. 11. 14. 19. May. 24 May. 28 Iun. 7 Iun. 14 Nov. 29 E 5. 16. Apr. 27 May. 1 May. 11 May. 18 Nov. 30 2. 10. 13. 18. May. 4 May. 8 May. 18 May. 25 Nov. 30 1. 4. 7. 12. 15. May. 11 May. 15 May. 25 Iun. 1 Nov. 30 6. 9. 14. 17. May. 18 May. 22 Iun. 1 Iun. 8 Nov. 30 3. 8. 11. 19. May. 25 May. 29 Iun. 8 Iun. 15 Nov. 30 F 5. 16. Apr. 28 May. 2 May. 12 May. 19 Dec. 1 2. 7. 10. 13. 18. May. 5 May. 9 May. 19 May. 26 Dec. 1 1. 4. 12. 15. May. 12 May. 16 May. 26 Iun. 2 Dec. 1 3. 6. 9. 14. 17. May. 10 May. 23 Iun. 2 Iun. 9 Dec. 1 8. 11. 19. May. 26 May. 30 Iun. 9 Iun. 16 Dec. 1 G 5. 13. 16. Apr. 29 May. 3 May. 13 May. 20 Dec. 2 2. 7. 10. 18. May. 6 May. 10 May. 20 May. 27 Dec. 2 1. 4. 9. 12. 15. May. 13 May. 17 May. 27 Iun. 3 Dec. 2 3. 6. 14. 17. May. 20 May. 24 Iun. 3 Iun. 10 Dec. 2 8. 11. 19. May. 27 May. 31 Iun. 10 Iun. 17 Dec. 2
 Dom Letter Golden Number Easter Terme begins Easter Terme ends Trinity Terme begins Trinity Terme ends A 2. 5. 13. 16 Apr. 12 May. 8 May. 26 Iun. 14 7. 10. 15. 18 Apr. 19 May. 15 Iun. 2 Iun. 21 1. 4. 9. 12 Apr. 26 May. 22 Iun. 9 Iun. 28 3. 6. 11. 14. 17 May. 3 May. 29 Iun. 16 Iuly. 5 B [...]. 5. 13. 16 Apr. 13 May. 9 May. 27 Iun. 15 4. 7. 10. 15. 18 Apr. 20 May. 16 Iun. 3 Iun. 22 1. 9. 12. 17 Apr. 27 May. 23 Iun. 10 Iun. 29 3. 6. 11. 14 May. 4 May. 3 Iun. 17 Iuly. 6 8 19 May. 11 Iun. 6 Iun. 24 Iuly. 13 C 2. 5. 10. 13. 16 Apr. 14 May. 10 May. 28 Iun. 16 4. 7. 15. 18 Apr. 21 May. 17 Iun. 4 Iun. 23 1. 6. 9. 12. 7 Apr. 28 May. 24 Iun. 11 Iun. 30 3. 11. 14. 19 May. 5 May. 31 Iun. 18 Iuly. 7 8 May. 12 Iun. 7 Iun. 25 Iuly. 14 D 16 Apr. 8 May. 4 May. 22 Iun. 10 2. 5. 10. 13 Apr. 15 May. 11 May. 29 Iun. 17 4. 7. 12. 15. 18 Apr. 22 May. 18 Iun. 5 Iun. 24 1. 6. 9. 17 Apr. 29 May. 25 Iun. 12 Iuly. 1 3. 8. 11. 14. 19 May 6 Iun. 1 Iun. 19 Iuly. 8 E 5. 16. Apr. 9 May. 5 May. 23 Iun. 11 2. 10. 13. 18 Apr. 16 May. 12 May. 30 Iun. 18 1. 4. 7. 12. 15. Apr. 23 May. 19 Iun. 6 Iun. 25 6. 9. 14. 17. Apr. 30 May. 26 Iun. 13 Iuly. 2 3. 8. 11. 19. May. 7 Iun. 2 Iun. 20 Iuly. 9 F 5. 16. Apr. 10 May. 6 May. 24 Iun. 12 2. 7. 10. 13. 18 Apr. 17 May. 13 May. 31 Iun. 19 1. 4. 12. 15. Apr. 24 May. 20 Iun. 7 Iun. 26 3. 6. 9. 14. 17. May. 1 May. 27 Iun. 14 Iuly. 3 8. 11. 19. May. 8 Iun. 3 Iun. 21 Iuly. 10 G 5. 13. 16. Apr. 11 May. 7 May. 25 Iun. 13 2. 7. 10. 18 Apr. 18 May. 14 Iun. 1 Iun. 20 1. 4. 9. 12. 15. Apr. 2 [...] May. 21 Iun. 8 Iun. 27 3. 6. 14. 17. May. 2 May. 28 Iun. 15 Iuly. 4 8. 11. 19. May. 9 Iun. 4 Iun. 22 Iuly. 11

Cambridg Commencment First Sunday in July. Oxon Act 2d. Sunday in July.

The Returnes of The Act of

• Trin. Terme are j. Crast: Trin 2 Oct: Trin: 3 Quind Trin: 4 Tivs. Trin
• Easter Terme. are. j. Quind. Pasch: 2 Tres Pa. 3. Mens Pa. 4 Quind: Pa. 5: Crō. Asc:
• Batchelors in Cambr: & Oxf: ye 1st. day of Lent and of Masters in
• Cambr. Iuly. 3
• Oxford. Iuly. 8
A Table of the Fixed Feasts, and other Solemn Days to be observed in the Church of England.
• All Sundays.
• NEw-years-day, or Circumcision Jan. 1
• Twelf day, or Epiphany Jan. 5
• Martyrdom of K. Charles I. Jan. 30
• Purification of the Virgin Mary Febru. 2
• LADY-DAY, or the Annuntiation of the Virgin Mary Mar. 25
• Mark Evang. April 25
• May day, or Phillip and Jacob May 1
• Birth and return of Charles II. May 29
• MIDSUMMER or John Baptist. June 24
• James Apostle July 25
• Bartholomew Apostle Aug. 24
• Matthew Apostle Sept. 21
• MICHAELMAS, or Mich. Archangel Sept. 29
• Luke the Evangelist Osto. 18
• Simon and Jude Octo. 28
• All Saints Nov. 1
• Powder Treason Nov. 5
• Andrew Apostle Nov. 30
• Thomas Apostle Dec. 21
• CHRISTMAS, or Birth of our Lord Dec. 25
• St. Stephen Dec. 26
• St. John Evang. Dec. 27
• Innocents Dec. 28
• [Page 16]
Other Remarkable Days.
• VAlentine Febr. 14
• Equal day and night Mar. 10
• St. George April 23
• Longest day, or Barneby June 11
• Swithin July 15
• Lammas Augu. 2
• Equal day and night Sept. 12
• Shortest day Dece. 11
• HILLARY TERM.
• BEgins Jan. 23
• Ends and hath Four Returns. Febr. 12
• 1. Octab. Hil. Jan. 20
• 2. Quind. Hil. Jan. 29
• 3. Crast Pur. Febr. 3
• 4. Octab. Pur. Febr. 10
• MICHAELMAS TERM
• Begins Octob. 23
• Ends and hath Six Returns. Nov. 28
• 1. Tres Mich. Octo. 21
• 2. Mens. Mich. Octo. 29
• 3. Crast. An. Nov. 4
• 4. Crast. Mar. Nov. 11
• 5. Oct. Mar. Nov. 18
• 6. Quin. Mar. Nov. 27
 Gol: Num D: of ye Mon. A Table for the ready finding what Sign the Moon is in or shall be for euer And what part of Mans body every Sign doth govern Febr: Nov: 3 1 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 ♈ Head & Face March 2 1 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 14 3 2 1 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 ♉ Neck & Throat December 6 4 3 2 1 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 5 4 3 2 1 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 ♊ Arms shoulders and Hands Aprill 17 6 5 4 3 2 1 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 9 7 6 5 4 3 2 1 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 ♋ Breast and stomack May 1 8 7 6 5 4 3 2 1 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 9 8 7 6 5 4 3 2 1 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 12 10 9 8 7 6 5 4 3 2 1 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 ♌ Heart and Back 4 11 10 9 8 7 6 5 4 3 2 1 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 Iune 12 11 10 9 8 7 6 5 4 3 2 1 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 ♍ Bowels & Belly 15 13 12 11 10 9 8 7 6 5 4 3 2 1 27 26 25 24 23 22 21 20 19 18 17 16 15 14 Iuly 7 14 13 12 11 10 9 8 7 6 5 4 3 2 1 27 26 25 24 23 22 21 20 19 18 17 16 15 ♎ Reyns & Loyns 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 27 26 25 24 23 22 21 20 19 18 17 16 18 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 27 26 25 24 23 22 21 20 19 18 17 ♏ Secrets 10 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 27 26 25 24 23 22 21 20 19 18 August 2 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 27 26 25 24 23 22 21 20 19 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 27 26 25 24 23 22 21 20 ♐ Thighs 13 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 27 26 25 24 23 22 21 5 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 27 26 25 24 23 22 ♑ Knees September 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 27 26 25 24 23 16 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 27 26 25 24 ♒ Leggs Ian: Oct: 8 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 27 26 25 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 27 26 ♓ Feet 19 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1 27 11 27 26 25 24 23 22 21 20 19 18 17 16 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

### The Ʋse of this Table.

1. SEek the name of the Month in the left-hand Margent, and guiding your Eye to (1) in the Table, find out the day of the Month, which you shall find either above, or beneath in that Column.

2. From that day of the Month, guide your eye back to the Number in the left-hand Margent that stands against it in the Column under the Title (Day of the Month.)

3. From that Number guide your Eye to (1) in the Table, and in that Column find the Number that is the Number of the Prime for that Year, and from thence guide your Eye to the right-hand Margent, so have you your desire.

Example.

The Tenth day of May, 1665. I desire to know what Sign the Moon is, &c.

1. The Prime for that Year is 13.

2. I find May in the left-hand Margent, and guiding my Eye to (1) in the Table, and in that Column to 10 the Day of the Month, I bring my Eye back to (17) in the left-hand Margent.

3. I look for 13 among the Primes, and from that guide my Eye to (1) in the Table, and finding the aforesaid Number (17 in that Column, I do from that, guide my Eye to the right-hand Mar­gent, and find that the Moon upon the Tenth of May, 1665. is entring into Leo Ω, and governs the Bowels and Belly.

### A Table shewing: the time of the Moons com­ing to the South, and quantity of her shining.

 The Moons age. Moons southing & shin. Moons age for her shi. 1 16 0 48 1 29 2 17 1 36 2 28 3 18 2 24 3 27 4 19 3 12 4 26 5 20 4 00 5 15 6 21 4 48 6 24 7 22 5 36 7 23 8 23 6 24 8 22 9 24 7 12 9 21 10 25 8 00 10 20 11 26 8 48 11 19 12 27 9 36 12 18 13 28 10 24 13 17 14 29 11 12 14 16 15 30 12 00 15

The use of this Table.

FInd the Moons age in the first Column, and next against the same towards the right hand, is the time of her coming to the South; which from the New Moon to the Full Moon, is always in the Afternoon, but from the Full to the New, it is in the Morning.

Example.

May 12. 1671. the Moon is fourteen days old, which I find in the first Column, against which, towards the right hand in the second Column is 11. 12, which being before the Full of the Moon, I conclude that the Moon comes to south May 12. 1671. at 11 a clock at Night, and 12 Minutes past.

[Page] To know how long the Moon Shineth.

Enter the third Column with the Moons age, and against it, on the left hand, you have the time of her shining, which all the time of her Encrease, being added to the hour of Sun rising, gives the time of her rising: But if added to the time of Sun setting, gives the time of her setting.

But after the Full.

Take the time of her shining from the Suns ri­sing, and it gives her rising; and then take the same from the Sun setting, it gives the time of her setting.

Example.

May 12. 1671. the Moon is 14 days old, and I find 11 hours, 12 minutes, for the time of her shining, (which being added to the Suns rising) upon the twelf of May, 1671. (viz. four hours) makes 3 of the clock, 12 minutes, for the time of the Moons rising the next Morning.

Again, to the said 11 hours, 12 minutes, add 8 hours from the Sun setting, it gives 7 hours, 12 minutes for the time of his setting.

Though these Rules are not altogether exact, yet they come near enough the truth, for ordinary use.

### A Tide-Table of certain Havens in and about Eng­land, whereby may be known what Moon makes a Full-Sea in any of the said places; and how many ho. and min. are to be added to the time of the Moons coming to the South for the time of High-water.

 South and North Queenborough Southamton, Ports­mouth, Isle of Wight, Spits, Ken­tish Knock, half-Tide at Dunkirk 0. H. 0 M. S by W N by E Rochester, Maldon, Aberdeen, Redband, West-end of the Nowr-Blacktail. 0 H. 45 M. S S W N N E Gravesend, Downs, Rumney, Ten net, Silly halfe tide, Blackness, Ramkines, Senebead. 1 H. 30 M. S W by S N E by N Dundee, St. Andrews, Lisborn St. Lucas, Bell Isle, Holy Isle. 2 H. 15 M. S W N E London, Tinmouth, Hartlepool Whitebay, Amsterdam, Gascoign, Britain, Galicia. 3 H. 0 M. S W by W N E by E Barwick, Hambrough-head, Brid­lington bay, Burdeux, Ostend, Flushing, Fountness. 3 H. 45 M. W S W E N E Scarborough quarter-tide, Lawre nas, Severn, Horkhave, Dungarum, Mounts-bay, Kingsale, Calice-Creek 4 H. 30 M. W by S E by N Newcastle, Humber, Falmouth Sal ly, Dartmouth, To bay, St. Mal­lows, Foy, Garsy. Liz. 5 H. 15 M. East and West Plimouth, Weymouth, Hull, Lyn, Davids head, Antwerp, Lundy, Holms of Bristol. 6 H. 0 M. E by S W b N Bristol, Foulness at the Start. 6 H. 45 M. [Page]E S E W N W Milford, Bridgewater, Lands-end, Waterford, Abermorick, Cape-Cleer, Texel. 7 H. 30 M. S E by E N W by W Portland, Peterport, Harflew, the Hague, S. Magnes. south, Dublin, Lambay. Macknels Cape. 8 H. 15 M. S E N W Pool, S. Hellen, Catnes, Orkney, Fair-Isles, Kilden, Man-Isle, Bass-Islands. 9 H. 0 M. S E by S N W by N Needles, Laisto, North & South Foreland. 9 H. 45 M. S S E N N W Tarmouth, Dover, Harwich, S. John de Luce, Calice Road, Bullein. 10 H. 30 M. S by E N by W Rye, Winchelsey, Goree, Thames, Rhodes. 11 H. 15 M.

### The Use of the Tide-Table.

Example.

May 12. 1671. I would know the Full Sea at London.

1. By the foregoing Rules I find the Moon comes to South at 11 of the clock, and 12 Minutes past, at Night.

There I seek for London in this Table, where I find that a S. W. or N. E. Moon makes a Full Sea, and on the right-hand I find 3 hours 0 minutes, which must be added to the Moons Southing. That is 3 hours, 0 minutes, added to 11 ho. 12 min. makes 2 a clock and 12 minutes the next Morning, for High Water at London-Bridge.

So that for any place and day, the hours and min. in the Table, are to be added to the Moons Southing, which gives the true time of High­water for that place and day.

### The time of the Suns Rising and Setting throughout the whole Year.

 Days of the Month. January February March Sun rises. Sun sets. Sun rises. Sun sets. Sun rises. Sun sets. H. M H M H. M H. M H. M. H. M. 1 8 9 3 51 7 18 4 42 6 20 5 40 2 8 8 3 51 7 17 4 43 6 18 5 42 3 8 7 3 53 7 15 4 45 6 16 5 44 4 8 6 3 54 7 13 4 47 6 14 5 46 5 8 4 3 55 7 11 4 49 6 11 5 49 6 8 2 3 56 7 9 4 51 6 8 5 51 7 8 1 3 58 7 7 4 58 6 6 5 54 8 8 0 4 0 7 5 4 55 6 4 5 56 9 7 58 4 1 7 3 4 57 6 2 5 58 10 7 56 4 3 7 1 4 59 6 0 6 0 11 7 55 4 5 6 59 5 1 5 58 6 2 12 7 54 4 6 6 57 5 3 5 56 6 4 13 7 52 4 8 6 54 5 6 5 54 6 6 14 7 51 4 9 6 52 5 8 5 52 6 8 15 7 50 4 10 6 50 5 10 5 50 6 10 16 7 48 4 12 6 48 5 12 5 48 6 12 17 7 46 4 14 6 46 5 14 5 46 6 14 18 7 44 4 16 6 44 5 16 5 44 6 16 19 7 43 4 17 6 42 5 18 5 42 6 18 20 7 41 4 19 6 40 5 20 5 40 6 20 21 7 39 4 21 6 38 5 22 5 38 6 22 22 7 37 4 23 6 36 5 24 5 36 6 24 23 7 35 4 24 6 34 5 26 5 34 6 26 24 7 34 4 26 6 32 5 28 5 32 6 28 25 7 32 4 28 6 30 5 30 5 30 6 30 26 7 30 4 30 6 29 5 31 5 28 6 32 27 7 28 4 31 6 27 5 33 5 26 6 34 28 7 26 4 34 6 24 5 36 5 24 6 36 29 7 24 4 36 6 22 5 22 6 38 30 7 22 4 38 5 20 6 40 31 7 20 4 40 5 18 6 42
 Days of the Month. April. May. June. Sun rises. Sun sets. Sun rises. Sun sets. Sun rises. Sun sets. H. M. H M. H. M. H. M. H. M H. M. 1 5 16 6 44 4 19 7 42 3 42 8 18 2 5 14 6 46 4 17 7 43 3 42 8 18 3 5 12 6 48 4 15 7 45 3 42 8 18 4 5 10 6 50 4 14 7 46 3 41 8 19 5 5 8 6 52 4 12 7 48 3 41 8 19 6 5 6 6 54 4 11 7 49 3 41 8 19 7 5 5 6 55 4 10 7 50 3 41 8 19 8 5 3 6 57 4 8 7 52 3 41 8 19 9 5 2 6 58 4 7 7 53 3 41 8 19 10 5 0 7 0 4 6 7 54 3 41 8 19 11 4 57 7 3 4 4 7 56 3 41 8 19 12 4 55 7 5 4 3 7 57 3 41 8 19 13 4 53 7 7 4 2 7 58 3 41 8 19 14 4 51 7 9 4 0 8 0 3 41 8 19 15 4 49 7 11 3 59 8 1 3 41 8 19 16 4 47 7 13 3 58 8 2 3 41 8 19 17 4 45 7 15 3 57 8 3 3 41 8 19 18 4 43 7 17 3 56 8 4 3 41 8 19 19 4 41 7 19 3 55 8 5 3 42 8 18 20 4 39 7 21 3 53 8 7 3 42 8 18 21 4 38 7 22 3 52 8 8 3 42 8 18 22 4 36 7 24 3 51 8 9 3 43 8 17 23 4 34 7 26 3 50 8 10 3 44 8 16 24 4 32 7 28 3 49 8 11 3 44 8 16 25 4 30 7 30 3 48 8 12 3 45 8 15 26 4 28 7 32 3 47 8 13 3 46 8 14 27 4 27 7 33 3 46 8 14 3 47 8 13 28 4 25 7 35 3 45 8 15 3 48 8 12 29 4 23 7 37 3 45 8 15 3 49 8 11 30 4 21 7 39 3 44 8 16 3 50 8 10 31 3 43 8 17
 Days of the Month. July. August. September Sun rises. Sun sets. Sun rises. Sun sets. Sun rises. Sun sets. H. M. H. M. H. M. H. M. H. M. H. M. 1 3 51 8 9 4 35 7 26 5 35 6 25 2 3 51 8 8 4 36 7 24 5 37 6 23 3 3 52 8 8 4 38 7 22 5 39 6 21 4 3 53 8 7 4 40 7 20 5 41 6 19 5 3 54 8 6 4 41 7 19 5 43 6 17 6 3 56 8 4 4 43 7 17 5 45 6 15 7 3 57 8 3 4 45 7 15 5 47 6 13 8 3 58 8 2 4 47 7 13 5 49 6 11 9 3 59 8 1 4 48 7 12 5 51 6 9 10 4 0 8 0 4 50 7 10 5 53 6 7 11 4 2 7 58 4 52 7 8 5 56 6 4 12 4 3 7 57 4 54 7 6 5 58 6 2 13 4 4 7 56 4 56 7 4 6 0 6 0 14 4 6 7 54 4 58 7 2 6 2 5 58 15 4 7 7 53 5 0 7 0 6 4 5 56 16 4 8 7 52 5 2 6 58 6 6 5 54 17 4 10 7 50 5 4 6 56 6 9 5 51 18 4 11 7 49 5 6 6 54 1 11 5 49 19 4 12 7 48 5 8 6 52 6 13 5 47 20 4 14 7 46 5 10 6 50 6 15 5 45 21 4 15 7 45 5 12 6 48 6 17 5 43 22 4 17 7 43 5 14 6 46 6 19 5 41 23 4 18 7 42 5 16 6 44 6 21 5 39 24 4 20 7 40 5 18 6 42 6 23 5 37 25 4 22 7 38 5 20 6 40 6 25 5 35 26 4 23 7 37 5 22 6 38 6 27 5 33 27 4 25 7 35 5 24 6 36 6 29 5 31 28 4 27 7 33 5 26 6 34 6 31 5 29 29 4 28 7 32 5 28 6 32 6 33 5 27 30 4 30 7 30 5 30 6 30 6 35 5 25 31 4 32 7 28 5 32 6 28
 Days of the Month. October. Novemb. December. Sun rises. Sun sets. Sun rises. Sun sets. Sun rises. Sun sets. H. M. H. M. H. M. H. M. H. M. H. M. 1 6 38 5 22 7 38 4 22 8 16 3 44 2 6 40 5 20 7 39 4 21 8 16 3 44 3 6 42 5 18 7 41 4 19 8 17 3 43 4 6 44 5 16 7 42 4 18 8 17 3 43 5 6 46 5 14 7 44 4 16 8 18 3 42 6 6 48 5 12 7 46 4 14 8 18 3 42 7 6 50 5 10 7 47 4 13 8 18 3 42 8 6 52 5 8 7 49 4 11 8 19 3 41 9 6 54 5 6 7 51 4 9 8 19 3 41 10 6 56 5 4 7 52 4 8 8 19 3 41 11 6 58 5 2 7 54 4 6 8 19 3 41 12 7 0 5 0 7 55 4 5 8 19 3 41 13 7 2 4 58 7 57 4 3 8 19 3 41 14 7 4 4 56 7 58 4 2 8 19 3 41 15 7 6 4 54 8 0 4 0 8 19 3 41 16 7 8 4 52 8 1 3 59 8 18 3 42 17 7 10 4 50 8 3 3 57 8 18 3 42 18 7 12 4 48 8 4 3 56 8 18 3 42 19 7 14 4 46 8 5 3 55 8 17 3 43 20 7 16 4 44 8 7 3 53 8 17 3 43 21 7 18 4 42 8 8 3 52 8 16 3 44 22 7 20 4 40 8 9 3 51 8 16 3 44 23 7 22 4 38 8 10 3 50 8 15 3 45 24 7 23 4 37 8 11 3 49 8 14 3 46 25 7 25 4 35 8 12 3 48 8 14 3 46 26 7 27 4 33 8 12 3 48 8 13 3 47 27 7 29 4 31 8 13 3 47 8 12 3 48 28 7 31 4 29 8 14 3 46 8 12 3 48 29 7 33 4 27 8 14 3 46 8 11 3 49 30 7 34 4 26 8 15 3 45 8 10 3 50 31 7 36 4 24 8 9 3 51

### A Table shewing the length of the longest Artificial Day, in all places from the Equinoctial, to the Poles of the World.

 Heig. Long. day Heig. Long. day Heig. Longest day. Pole. H. M. Pole. H. M. Pole. D. H. M. 00 12 00 47 15 42 68 42 01 16 06 12 20 48 15 52 69 54 16 25 12 12 42 49 16 00 70 64 13 46 16 12 58 50 16 10 71 74 00 00 20 13 12 51 16 20 72 82 06 36 24 13 30 52 16 30 73 89 04 58 27 13 42 53 16 42 74 96 17 00 30 13 56 54 16 54 75 104 01 04 32 14 06 55 17 08 76 110 07 27 34 14 16 56 17 22 77 116 14 22 35 14 22 57 17 36 78 122 17 06 36 14 28 58 17 52 79 127 09 55 37 14 34 59 18 10 80 134 04 58 38 14 38 60 18 30 81 139 31 36 39 14 44 61 18 54 82 145 06 43 40 14 52 62 19 20 83 152 02 06 41 14 58 63 19 50 84 156 03 03 42 15 04 64 20 24 85 161 05 23 43 15 12 65 21 10 86 166 11 23 44 15 18 66 22 18 87 171 21 47 45 15 26 66½ 24 00 88 176 05 29 46 15 34 67 24 Days 89 181 21 58 90 187 06 39

[Page] The following TABLES Are of excellent use, and do readily discover the exact time of the New Moon, Full Moon, As likewise the First and Second Quadrats; And consequently her true Age.

And this from the year of our Lord 1673, to the year 1700.

EXAMPLE.

In the Moneth of April 1673, and the 14th day of the Moneth, the Table for that year, will discover, over against the said Moneth, April

First That the New Moon happens to be the fixth day of that Moneth, and the 13th hour of that day; That is, 10 Minutes past 1 of the Clock at night (remembring always that the dayes are to be accompted from Noon.)

Secondly. That the first Quadrat is the 13th day, 10 min. past 9 at night.

Thirdly. That the Full Moon is the 20th day, 11 min. past 12 at night.

Fourthly. The Second Quadrat is the 28th day, 1 Min. past 12 at night.

Fifthly, And Lastly, because the Moon changes on the sixth day, and 8 added to 6 makes 14, therefore the Moon is 8 dayes old, upon the said 14th day of April.

But if you will be more exact, you must Accompt

 For the First Quarter of the Moon 7 d. 09h. 11 m Full Moon 14 d. 18 h. 22 m Last Quart. 22 d. 03 h. 33 m Time from Moon to Moon 29 d. 12 h. 44 m

An Explanation of the double Numbers in the Table.

EXAMPLE.

In the first Quarter of the Moon, in the Moneth of May, in the year 1674. I find the 2 Numbers, viz.

[...]

The meaning whereof is, that in the said moneth of May, the first quarter of the Moon happeneth to be both upon the first day, 12th hour, 45th minute, And likewise upon the 3 [...]th day, 16 houres, 30 m. of the same Moneth. The which is to be so read, and so understood in any other year or month.

 1673. New ☽ 1. Quar. Full ☽ 2. Quar. D. H. M. D. H. M. D. H. M. D. H. M. January 7 14 07 15 15 37 22 01 10 29 05 20 Februa. 6 06 29 13 20 58 20 12 06 27 23 11 March 8 00 12 15 04 06 21 23 38 29 18 12 April 6 13 10 13 09 14 20 12 11 28 12 1 May 5 23 11 12 14 12 20 01 35 28 10 26 June 4 07 10 10 21 05 18 15 57 26 18 20 July 3 14 09 10 06 38 18 06 59 26 03 42 August 1 31 20 5 44 17 8 19 24 16 22 21 24 15 01 Septem. 29 15 15 7 11 17 15 13 42 22 22 22 October 29 3 58 7 5 51 15 4 25 22 4 41 Novem. 27 19 37 6 1 59 13 18 6 20 11 47 Decem. 27 9 17 5 22 33 13 6 19 19 20 59

 1674. New ☽ 1. Quar. Full ☽ 2. Quar. D. H. M. D. H. M. D. H. M. D. H. M. January 26 08 41 4 16 17 11 17 22 18 09 09 Februa. 15 03 15 3 09 51 10 03 15 16 23 47 March 26 18 52 4 22 44 11 12 45 18 16 12 April 25 09 00 3 07 19 9 22 16 17 09 25 May 24 21 01 1 31 12 16 45 30 9 06 27 17 02 54 June 23 06 24 29 21 07 7 19 42 15 19 44 July 22 14 22 29 04 24 7 08 40 5 11 32 August 20 21 58 27 14 47 5 23 12 14 1 56 Septem. 19 0 5 26 5 0 4 15 49 12 14 25 October 18 15 40 25 22 41 4 8 18 12 1 9 Novem. 17 3 7 24 19 3 3 1 18 10 9 45 Decem. 16 17 14 24 15 54 2 17 16 9 17 14
 1675. New ☽ 1. Quar. Full ☽ 2. Quar. D. H. M. D. H. M. D. H. M. D. H. M. January 15 9 36 23 12 26 1 7 22 8 6 23 Februa. 14 3 4 22 6 38 No Full ☽ 6 9 57 March 15 20 50 23 21 27 1 30 5 18 17 54 7 20 53 April 14 13 28 22 8 15 28 23 41 6 10 34 May 14 4 23 21 14 21 28 6 18 6 1 16 June 12 17 16 19 19 49 26 15 34 4 17 21 July 12 4 16 18 23 31 26 2 47 4 10 11 August 10 15 46 17 4 39 24 16 25 3 3 17 Septem. 8 22 54 15 12 44 23 8 30 1 19 57 October 8 7 47 15 1 11 23 2 22 1 31 11 1 39 47 Novem. 6 17 24 13 17 6 21 21 8 29 13 27 Decem. 6 4 10 13 12 13 21 15 2 28 22 45

 1676. New ☽ 1. Quar. Full ☽ 2. Quar. D. H. M. D. H. M. D. H. M. D. H. M. January 4 26 38 12 8 57 20 6 58 27 6 8 Februa. 3 6 39 11 4 49 18 21 56 25 12 58 March 3 22 7 12 0 43 19 7 9 25 20 50 April 2 14 11 10 17 4 17 14 59 24 6 40 May 2 31 6 21 8 16 10 6 7 16 22 17 23 18 29 June 13 11 18 8 17 36 15 5 26 22 8 44 July 30 0 37 7 22 13 14 13 23 22 0 33 August 28 9 32 6 2 59 12 23 2 20 18 10 Septem. 26 23 16 4 7 48 1 11 32 19 12 34 October 26 9 39 3 14 15 11 2 39 19 6 49 Novem. 24 19 52 2 0 6 9 20 49 18 0 2 Decem. 24 6 14 1 31 14 6 4 57 9 16 4 17 15 5
 1677. New ☽ 1. Quar. Full ☽ 2. Quar. D. H. M. D. H. M. D. H. M. D. H. M January 22 17 30 30 3 1 8 11 16 16 2 39 Februa. 21 5 27 28 21 51 7 14 17 14 11 11 March 22 14 5 30 17 59 8 19 8 15 17 19 April 21 7 53 29 11 10 7 6 30 13 22 30 May 20 22 34 29 1 9 6 15 25 13 5 20 June 19 13 15 27 13 13 4 22 45 11 14 11 July 9 3 35 16 22 56 4 5 30 11 1 46 August 17 19 5 25 4 12 2 31 9 0 18 25 9 16 22 Septem. 16 9 18 23 11 57 30 8 44 8 9 43 October 15 23 2 22 17 51 29 22 49 8 4 37 Novem. 14 11 5 21 2 12 28 13 46 7 0 23 Decem. 13 22 8 20 13 23 28 10 56 6 16 1

 1678. New ☽ 1. Quar. Full ☽ 2. Quar. D. H. M. D. H. M. D. H. M. D. H. M. January 12 5 21 19 3 36 27 6 14 5 13 43 Februa. 10 18 52 17 19 58 26 0 20 4 4 22 March 12 5 24 19 12 34 27 15 51 5 14 33 April 10 16 28 18 8 8 26 4 30 3 21 32 May 10 2 50 18 1 29 25 14 41 3 2 2 June 8 15 25 16 17 44 23 22 40 1 30 6 12 22 35 July 8 9 52 16 8 18 23 6 2 29 21 25 August 6 20 31 14 20 57 21 13 26 28 10 2 Septem. 5 12 41 13 7 30 19 21 52 27 1 46 October 5 5 7 12 16 4 19 8 8 26 20 52 Novem. 3 21 2 10 23 20 17 20 53 25 17 30 Decem. 3 11 24 10 6 39 17 12 25 25 14 33
 1679. New ☽ 1. Quar. Full ☽ 2. Quar. D. H. M. D. H. M. D. H. M. D. H. M. January 2 0 17 8 15 20 16 6 1 24 9 58 Februa. No New ☽ 7 2 26 15 0 28 23 2 26 March 1 31 20 5 48 32 8 15 47 16 16 8 24 15 15 April 29 7 26 7 6 56 14 23 45 22 23 57 May 28 23 25 6 23 4 15 0 23 22 12 33 June 27 9 59 5 15 58 13 15 54 20 9 8 July 26 22 31 5 8 51 12 22 43 19 13 12 August 25 13 14 4 1 6 11 6 15 17 20 3 Septem. 24 5 55 2 16 52 9 14 35 16 6 17 October 24 3 31 2 31 6 18 39 38 8 23 16 15 20 39 Novem. 22 15 43 30 4 13 7 9 10 14 14 20 Decem. 22 10 26 29 12 20 6 21 9 14 10 35

 1680. New ☽ 1. Quar. Full ☽ 2. Quar. D. H. M. D. H. M. D. H. M. D. H. M. January 21 0 59 27 19 19 5 10 6 13 7 3 Februa. 19 12 57 26 3 36 4 2 1 12 03 58 March 19 22 20 26 11 26 4 18 39 12 21 55 April 18 6 17 25 1 52 3 11 25 11 12 54 May 17 13 50 24 14 59 3 3 9 11 2 56 June 15 21 31 23 6 17 1 18 13 9 7 20 July 15 16 39 22 23 12 1 30 39 19 22 17 8 12 15 August 13 7 32 21 16 47 29 5 43 6 15 2 Septem. 12 6 57 20 10 31 27 8 8 4 21 28 October 13 2 58 20 1 30 27 1 19 4 5 23 Novem. 10 18 13 18 18 50 25 11 20 2 16 19 Decem. 10 13 33 18 7 34 24 20 22 2 9 34
 1681. New ☽ 1. Quar. Full ☽ 2. Quar. D. H. M. D. H. M. D. H. M. D. H. M. January 9 7 52 16 17 25 23 10 6 1 31 4 0 18 27 Februa. 7 23 48 15 0 37 21 23 6 No 2. Qu. March 9 12 43 16 6 45 23 13 24 1 31 20 15 45 15 April 7 23 3 14 13 24 22 4 29 30 7 34 May 7 7 11 13 21 47 21 19 58 29 20 41 June 5 13 7 12 8 53 20 10 45 28 6 24 July 4 21 41 11 22 40 20 1 18 27 14 25 August 3 4 57 10 14 3 18 14 20 25 20 7 Septem. 1 14 43 9 8 9 17 4 5 24 1 3 October 1 30 3 18 9 43 9 3 20 16 16 4 23 8 1 Novem. 29 12 56 7 22 38 15 5 8 21 17 50 Decem. 29 8 26 7 16 36 14 13 52 21 8 8

 1682. New ☽ 1. Quar. Full ☽ 2. Quar. D. H. M. D. H. M. D. H. M. D. H. M. January 28 3 32 6 8 21 13 0 12 19 23 20 Februa. 26 20 48 4 20 58 11 11 25 18 17 37 March 28 10 56 6 5 11 12 21 45 20 12 33 April 26 22 9 4 10 51 11 9 34 19 6 45 May 26 7 2 3 15 29 10 22 17 18 23 49 June 24 14 52 1 20 49 9 11 57 17 14 50 July 23 21 31 1 30 5 14 4 13 9 2 32 17 3 57 August 22 4 44 26 6 31 7 17 6 15 14 30 Septem. 20 13 59 28 0 2 6 9 34 13 23 0 October 20 1 18 27 19 39 6 1 4 13 5 58 Novem. 18 15 22 26 16 17 4 15 3 11 14 53 Decem. 18 8 32 26 12 20 4 5 7 10 20 37
 1683. New ☽ 1. Quar. Full ☽ 2. Quar. D. H. M. D. H. M. D. H. M. D. H. M. January 17 3 10 25 6 22 2 18 28 9 7 5 Februa. 15 21 57 23 21 6 1 2 39 7 20 17 March 17 15 39 25 7 36 2 31 7 21 58 45 9 11 32 April 16 6 46 23 14 28 30 7 5 8 4 15 May 15 19 26 22 18 37 29 17 40 7 21 32 June 14 5 40 20 22 31 28 5 12 6 14 34 July 13 14 15 20 5 40 27 19 3 6 7 4 August 11 22 9 18 12 43 26 10 31 4 22 31 Septem. 10 5 55 17 1 7 25 3 23 3 12 9 October 9 14 55 16 17 17 24 20 52 3 0 8 Novem. 8 1 26 15 12 25 23 13 42 1 30 10 17 2 42 Decem. 7 14 16 15 9 21 23 5 2 30 1 34

 1684. New ☽ 2. Quar. Full ☽ 1. Quar. D. H. M. D. H. M. D. H. M. D. H. M. January 6 5 34 14 6 25 21 20 3 28 9 42 Februa. 4 22 16 13 1 51 20 4 49 26 19 30 March 5 15 25 13 18 13 20 14 0 27 7 29 April 4 9 1 12 7 8 18 21 55 25 21 10 May 4 0 51 11 16 3 18 5 34 25 12 32 June 2 14 42 9 21 50 16 14 25 24 5 16 July 2 31 2 14 26 7 9 1 37 16 0 31 23 21 52 August 29 22 20 7 7 35 14 12 52 22 15 4 Septem. 28 7 33 5 12 20 13 8 0 21 7 40 October 27 16 46 4 21 20 12 21 5 20 23 0 Novem. 26 2 58 3 14 11 11 15 37 19 11 18 Decem. 25 14 32 3 6 27 11 10 18 18 23 0
 1685. New ☽ 1. Quar. Full ☽ 2. Quar. D. H. M. D. H. M. D. H. M. D. H. M. January 24 3 38 2/31 2/23 28/4 10 3 34 17 7 5 Februa. 22 18 19 No 1. Qu. 8 17 52 15 13 59 March 24 9 45 2 20 4 10 5 49 16 20 58 April 23 1 35 1 9 44 8 14 58 15 5 25 1 3 29 May 22 18 1 3 [...] 14 57 7 22 48 14 15 59 June 21 7 49 282 3 10 6 5 0 13 4 40 July 20 21 22 28 4 50 5 13 4 12 19 41 August 19 10 10 26 9 19 3 21 41 11 12 37 Septem. 17 21 53 24 14 57 2 8 39 10 9 55 October 17 8 45 23 22 50 1/31 22/15 36/26 10 1 31 Novem. 15 20 16 22 10 52 30 10 24 8 19 42 Decem. 15 5 27 22 2 20 30 5 56 8 8 5

 1686. New ☽ 1. Quar. Full ☽ 2. Quar. D. H. M. D. H. M. D. H. M. D. H. M. January 13 11 59 20 20 50 28 22 12 7 1 12 Februa. 12 1 26 20 15 37 27 16 20 5 7 57 March 13 14 2 21 11 1 29 6 56 6 18 39 April 12 4 32 20 5 13 27 15 1 5 0 7 May 11 16 9 19 21 18 26 23 0 4 5 41 June 10 9 16 18 11 42 25 5 55 2 12 50 July 9 23 59 17 21 57 24 12 40 1 31 23 14 5 12 August 8 15 7 16 4 15 22 20 50 30 4 2 Septem. 7 5 58 14 13 15 21 6 46 28 2 32 October 6 20 1 13 19 18 20 19 22 28 18 17 Novem. 5 9 16 12 2 13 19 11 24 27 14 30 Decem. 4 21 4 11 13 5 19 5 15 27 9 21
 1687. New ☽ 1. Quar. Full ☽ 2. Quar. D. H. M. D. H. M. D. H. M. D. H. M. January 3 7 55 10 0 24 18 0 27 26 1 28 Februa. 1 18 13 8 12 38 16 19 26 22 22 47 March 3 4 3 10 8 49 18 12 17 25 22 47 April 1 14 8 9 2 25 17 2 18 24 4 2 May 1 30 0 12 40 24 8 19 55 16 14 4 23 8 12 June 29 1 29 7 12 56 14 22 30 21 12 52 July 28 16 4 7 4 51 14 6 24 20 20 12 August 27 7 59 5 18 38 12 15 11 19 2 56 Septem. 26 0 29 4 6 2 10 21 30 17 21 16 October 25 10 50 3 14 10 10 6 49 17 14 53 Novem. 24 8 38 2 0 22 8 18 26 16 10 56 Decem. 23 22 30 1 30 7 15 16 16 8 8 31 16 8 5

 1688. New ☽ 1. Quar. Full ☽ 2. Quar. D. H. M. D. H. M. D. H. M. D. H. M. January 22 10 12 29 0 57 7 1 12 15 4 50 Februa. 20 20 24 27 12 39 5 19 4 13 22 30 March 21 5 15 28 2 29 6 13 58 14 13 14 April 19 15 27 26 18 14 5 6 4 13 0 4 May 18 16 10 26 10 41 4 21 36 12 6 56 June 17 8 5 25 3 34 3 9 56 10 5 46 July 16 19 47 24 20 34 2 20 38 9 15 10 August 15 9 10 23 12 34 1 30 5 14 49 26 7 20 45 Septem. 14 0 49 22 3 42 28 23 0 6 4 28 October 13 18 40 21 16 59 28
 1689. New ☽ 1. Quar. Full ☽ 2. Quar. D. H. M. D. H. M. D. H. M. D. H. M. January 10 22 10 17 20 26 24 22 12 3 1 2 Februa. 9 11 21 26 3 41 23 14 12 1 11 56 March 10 21 57 17 12 17 25 16 4 3 17 8 April 9 6 32 15 22 42 23 23 0 2 9 31 May 8 13 34 15 11 21 23 14 19 1 31 22 7 26 46 June 6 22 8 14 1 51 22 4 18 29 13 56 July 6 5 28 17 13 54 21 16 58 28 18 7 August 4 15 21 12 11 13 20 4 17 26 22 38 Septem. 3 3 39 11 5 17 18 17 46 25 8 13 October 2 18 55 10 23 0 18 0 40 24 14 21 Novem. 1 12 35 9 15 35 16 10 30 23 5 22 Decem. 1 31 1 3 4 2 9 6 0 15 22 2 22 22 4 [...]

 1690. New ☽ 1. Quar. Full ☽ 2. Quar. D. H. M. D. H. M. D. H. M. D. H. M. January 29 20 17 7 16 53 14 15 38 21 18 8 Februa. 28 11 4 6 1 49 12 20 39 20 14 24 March 29 22 8 7 8 6 14 10 14 22 9 46 April 28 6 59 5 14 2 13 21 34 21 3 7 May 27 14 31 4 21 0 12 15 19 20 17 53 June 25 21 21 3 6 30 11 4 13 19 5 43 July 25 4 42 2 18 27 10 21 28 18 14 45 August 23 13 21 1 31 9 2 18 23 9 11 45 16 21 32 Septem. 22 0 32 29 21 12 8 1 42 15 3 8 October 21 13 10 29 15 19 7 14 26 14 10 35 Novem. 20 7 34 28 9 51 6 0 25 12 16 52 Decemb. 20 2 42 28 5 5 5 13 11 12 4 25
 1691. New ☽ 1. Quar. Full ☽ 2. Quar. D. H. M. D. H. M. D. H. M. D. H. M. January 18 22 8 26 19 10 3 23 40 10 18 55 Februa. 17 16 28 25 5 39 2 9 51 9 12 9 March 19 7 52 26 13 36 3 20 25 11 6 30 April 17 21 40 24 19 4 2 7 22 10 1 5 May 17 6 29 25 21 50 1 31 20 7 12 54 9 17 37 June 15 14 38 22 4 24 29 22 26 8 10 46 July 14 21 49 21 13 43 29 13 29 8 1 2 August 13 4 50 20 2 22 28 5 16 6 13 6 Septem. 11 13 13 18 18 13 26 21 8 4 23 0 October 11 1 4 18 13 9 26 12 29 4 6 42 Novem. 9 12 15 17 9 44 25 2 30 2 13 42 1 20 54 Decem. 9 1 51 17 6 33 24 15 9 31 4 29

 1692. New ☽ 1. Quar. Full ☽ 2. Quar. D. H. M. D. H. M. D. H. M. D. H. M. January 7 21 52 16 0 46 23 2 3 29 17 34 Februa. 6 16 31 14 18 26 21 11 48 28 7 37 March 7 10 30 15 1 42 21 20 53 28 23 36 April 6 3 6 13 15 33 20 5 49 27 16 6 May 5 16 48 12 20 48 19 15 40 27 9 9 June 4 4 28 11 0 37 18 2 22 26 2 4 July 2 16 25 10 5 14 17 15 9 25 17 51 August 1 31 21 5 48 49 8 11 44 16 5 55 24 9 5 Septem. 29 14 24 6 22 4 4 22 26 22 21 13 October 29 0 36 6 12 30 14 15 52 22 9 32 Novem. 27 11 48 5 6 46 13 9 46 20 18 43 Decem. 27 1 35 5 2 40 13 1 50 20 0 46
 1693. New ☽ 1. Quar. Full ☽ 2. Quar. D. H. M. D. H. M. D. H. M. D. H. M. January 25 17 38 4 0 1 11 16 0 18 9 55 Februa. 24 10 50 2 19 53 10 3 52 16 20 13 March 26 4 2 4 14 17 11 13 32 18 5 19 April 24 20 20 3 4 52 9 21 44 16 17 56 May 24 11 7 2 31 5 3 47 6 9 5 37 16 7 59 June 22 23 56 30 3 44 7 13 36 14 23 37 July 22 11 7 29 7 33 6 22 40 14 16 22 August 20 21 26 27 12 39 5 9 50 13 9 58 Septem. 19 6 57 25 1 16 3 23 34 12 1 0 October 18 16 17 25 9 14 3 15 41 11 19 25 Novem. 17 2 9 24 0 42 2 9 16 10 10 2 Decem. 16 11 17 23 20 3 2 31 5 23 16 24 9 22 28

 1694. New ☽ 1. Quar. Full ☽ 2. Quar. D. H. M. D. H. M. D. H. M. D. H. M. January 15 1 14 22 19 39 30 15 34 8 7 53 Februa. 13 14 50 21 13 1 No Full ☽ 6 15 7 March 15 5 41 23 7 52 1 30 4 14 13 31 7 21 40 April 13 21 6 22 0 5 28 22 13 6 4 50 May 13 12 41 21 13 11 28 5 56 5 14 2 June 12 3 22 19 23 22 26 11 51 4 1 28 July 11 18 6 16 6 1 25 20 50 3 15 9 August 10 7 27 17 11 12 24 6 30 2 7 4 Septem. 8 20 10 15 17 1 22 19 8 1 30 0 19 55 44 October 8 7 32 14 22 48 22 10 34 30 14 36 Novem. 6 18 20 13 8 46 21 4 42 29 8 14 Decem. 6 4 43 12 22 20 21 0 8 28 23 31
 1695. New ☽ 1. Quar. Full ☽ 2. Quar. D. H. M. D. H. M. D. H. M. D. H. M. January 4 15 5 11 14 59 19 17 41 27 13 23 Februa. 3 1 52 10 9 40 18 12 13 25 20 1 March 4 13 21 12 5 3 20 2 50 27 1 46 April 3 1 41 10 23 46 18 13 57 25 7 11 May 2 14 59 10 16 54 17 23 8 24 12 53 June 1 30 4 1 [...] 56 41 9 7 56 16 6 17 22 21 26 July 30 10 52 8 20 20 15 13 10 22 8 42 August 29 2 13 7 6 20 13 20 33 20 23 10 Septem. 27 16 55 5 14 5 12 5 30 19 16 30 October 27 6 57 4 9 26 11 16 45 19 11 56 Novem. 25 16 7 3 3 10 10 7 0 18 10 37 Decem. 25 6 11 2 31 11 21 12 43 9 23 59 18 4 25

 1696. New ☽ 1. Quar. Full ☽ 2. Quar. D. H. M. D. H. M. D. H. M. D. H. M. January 23 17 36 30 11 49 8 18 36 16 21 59 Februa. 21 3 20 28 3 43 7 14 10 15 12 21 March 22 12 59 29 21 22 8 7 51 15 22 25 April 20 23 0 28 14 41 6 23 10 14 5 42 May 20 9 52 28 8 12 6 11 45 13 10 33 June 18 22 32 27 0 8 4 21 48 11 14 42 July 18 11 52 26 14 56 4 6 56 10 19 54 August 17 3 10 25 4 15 2 31 13 21 35 21 9 2 36 Septem. 15 19 46 23 15 29 30 5 52 7 17 17 October 15 11 36 23 0 52 29 16 30 7 9 23 Novem. 14 5 12 21 8 16 28 5 22 6 4 31 Decem. 13 20 4 20 15 49 27 20 49 6 0 42
 1697. New ☽ 1. Quar. Full ☽ 2. Quar. D. H. M. D. H. M. D. H. M. D. H. M. January 12 8 59 19 0 1 26 13 53 4 22 28 Februa. 10 21 34 17 10 30 25 8 10 3 17 43 March 12 5 14 18 23 24 27 1 23 5 10 20 April 10 13 32 17 13 52 25 17 27 3 23 0 May 9 21 39 17 5 37 25 7 13 3 7 46 June 8 6 36 15 22 16 23 18 59 1 30 11 17 10 2 July 7 16 58 15 15 2 23 5 1 29 21 10 August 6 5 27 14 8 1 21 13 54 28 3 48 Septem. 4 20 17 12 23 56 19 22 10 26 14 6 October 4 13 7 12 14 40 19 7 44 26 14 28 Novem. 3 7 46 11 3 13 17 17 51 24 22 14 Decem. 3 2 0 10 13 25 17 5 31 24 18 4

 1698. New ☽ 1. Quar. Full ☽ 2. Quar. D. H. M. D. H. M. D. H. M. D. H. M. January 1/31 20/13 48/16 8 21 24 15 19 7 23 15 26 Februa. No New ☽ 7 4 22 14 10 2 22 11 33 March 1 31 19 8 34 13 8 12 4 16 4 18 24 5 20 April 29 14 14 6 22 8 14 18 19 22 19 55 May 28 21 10 6 8 24 14 9 55 22 7 11 June 27 4 58 4 17 32 13 0 40 20 14 59 July 26 13 51 4 12 55 12 14 20 19 20 36 August 25 0 52 3 5 40 11 2 25 18 0 25 Septem. 23 14 29 1 23 36 9 13 30 16 5 1 October 23 7 24 1 31 71 11 53 26 9 0 2 15 14 3 Novem. 22 2 10 30 3 16 7 9 59 14 2 3 Decem. 21 21 37 29 16 0 6 19 54 13 17 48
 1699. New ☽ 1. Quar. Full ☽ 2. Quar. D. H. M. D. H. M. D. H. M. D. H. M. January 20 15 58 28 2 20 5 6 59 12 12 10 Februa. 19 7 52 26 9 42 3 14 49 11 7 58 March 20 20 54 27 15 13 5 7 14 13 3 45 April 19 6 46 25 1 4 3 20 50 11 21 44 May 18 14 43 25 5 7 3 11 13 11 14 10 June 16 21 32 23 11 45 2 0 12 10 3 16 July 16 4 38 23 4 53 1 31 17 7 22 52 9 14 27 August 14 12 32 21 20 49 29 22 22 7 22 10 Septem. 12 22 29 20 15 4 28 14 10 6 4 15 October 12 10 41 20 10 43 28 0 26 5 9 47 Novem. 11 2 43 19 6 37 26 12 3 3 8 25 Decem. 10 20 51 19 0 56 25 22 58 2 20 11

 1700. New ☽ 1. Quar. Full ☽ 2. Quar. D. H. M. D. H. M. D. H. M. D. H. M. January 9 16 26 17 14 3 24 9 4 1 31 16 7 27 14 Februa. 8 11 31 16 5 9 22 19 11 No 2. Qu March 10 4 18 17 13 34 24 5 39 2 0 50 31 19 33 April 8 18 18 15 19 3 22 17 0 30 13 22 May 8 5 32 14 23 22 21 5 6 30 6 6 June 6 14 26 13 4 34 20 18 25 28 21 10 July 5 21 42 12 12 13 20 9 2 28 11 0 August 4 5 7 10 23 10 19 0 42 26 22 2 Septem. 2 12 54 9 13 33 17 16 48 25 4 20 October 1 31 21 9 56 36 9 7 12 17 8 54 24 15 45 Novem. 29 23 50 8 3 10 16 0 1 22 21 38 Decem. 29 18 0 7 23 44 15 13 6 22 9 0
 Year. Moneth. D. H. M. Lun. Dig. Min 1674 January 11 17 22 ☽ 11 21 1674 July 7 18 39 ☽ 12 17 1675 January 1 7 29 ☽ 18 53 1675 June 26 15 30 ☽ 15 2 1676 May 31 21 20 ☉ 4 50 1677 May 6 15 25 ☽ 8 15 1678 October 19 8 17 ☽ 22 18 1681 August 18 14 22 ☽ 10 35 1682 August 7 17 56 ☽ 18 50 1682 February 11 11 28 ☽ 19 48 1683 January 17 4 27 ☉ 10 30 1684 June 16 14 17 ☽ 1 35 1684 July 2 3 10 ☉ 8 0 1685 November 30 10 26 ☽ 21 45 1686 November 19 11 22 ☽ 7 35 1687 May 1 1 25 ☉ 1 40 1688 April 5 6 4 ☽ 6 49 1689 March 25 6 41 ☽ 21 19 1689 September 18 14 46 ☽ 20 39 1690 March 14 10 14 ☽ 5 53 1692 July 17 15 9 ☽ 12 28 1693 February 11 16 24 ☽ 19 32 1693 June 23 0 6 ☉ 2 34 1694 June 26 12 50 ☽ 6 47 1695 November 10 7 0 ☽ 6 55 1696 May 6 11 45 ☽ 21 48 1696 October 29 16 30 ☽ 21 45 1697 October 19 7 44 ☽ 8 54 1699 March 5 7 14 ☽ 9 5 1699 September 12 21 30 ☉ 10 0

### A Table shewing the beginning of every Kings Reign, from the Conquest; Together with the year of Christ, answering to every year of each King or Queens Reign, from Henry 8. to Charles 2. inclusive. The year beginning on the 25th of March.

WILLIAM the Conquerer, BEgan his Reign the Fifteenth of October, 1066. Ended it the Ninth of September, 1087. Reigned 20 Years, 10 Months, 21 Days.

WILLIAM RƲEƲS. Began his Reign September the Ninth, 1087. Reigned 12 Years, 11 Months, 18 Days.

HENRY I. Began his Reign the First of August, 1100. Reigned 35 Years, 4 Months, 12 Days.

STEPHEN, Began his Reign December the Second, 1135. Reigned 18 Years, 11 Months, 20 Days.

HENRY II. Began his Reign October the Twenty Fifth, 1154. Reigned 34 Years, 9 Months, 5 Days.

RICHARD I. Began his Reign July the Ninth, 1189. Reigned 9 Years, 9 Months, 19 Days.

JOHN. Began his Reign April the Sixth, 1199. Reigned 17 Years, 7 Months, 0 Days.

[Page 2] HENRY III. Began his Reign October the Nineteenth, 1216. Reigned 56 Years, 1 Month, 0 Days.

EDWARD I. Began his Reign November the Sixteenth, 1272. Reigned 34 Years, 8 Months, 9 Days.

EDWARD II. Began his Reign July the Seventh, 1307. Reigned 19 Years, 7 Months, 9 Days.

EDWARD III. Began his Reign January the Twenty Fifth, 1326. Reigned 50 Years, 5 Months, 7 Days.

RICHARD II. Began his Reign June the Twenty First, 1377. Reigned 22 Years, 3 Months, 14 Days.

HENRY IV. Began his Reign September the Twenty Ninth, 1399. Reigned 13 Years, 6 Months, 3 Days.

HENRY V. Began his Reign March the Twentieth, 1412. Reigned 9 Years, 5 Months, 24 Days.

HENRY VI. Began his Reign August the Thirty First, 1422. Reigned 38 Years, 6 Months, 16 Days.

EDWARD IV. Began his Reign March the Fourth, 1460. Reigned 22 Years, 1 Month, 8 Days.

RICHARD III. Began his Reign June the Twenty Second, 1483. Reigned 2 Years, 2 Months, 5 Days.

HENRY VII. Began his Reign August the Twenty Second, 1485. Reigned 23 Years, 8 Months, 19 Days.

[Page 3] HENRY VIII. Began his Reign April the Twenty Second, 1509. Reigned 37 Years, 10 Months, 1 Day.

 Years of his Reign. Anno Dom. Years expired, Mar. 25. 1672. 1 1510 162 2 1511 161 3 1512 160 4 1513 159 5 1514 158 6 1515 157 7 1516 156 8 1517 155 9 1518 154 10 1519 153 11 1520 152 12 1521 151 13 1522 150 14 1523 149 15 1524 148 16 1525 147 17 1526 146 18 1527 145 19 1528 144 20 1529 143 21 1530 142 22 1531 141 23 1532 140 24 1533 139 25 1534 138 26 1535 137 27 1536 136 28 1537 135 29 1538 134 [Page 4]30 1539 133 31 1540 132 32 1541 131 33 1542 130 34 1543 129 35 1544 128 36 1545 127 37 1546 126

EDWARD VI. Began his Reign January the Twenty Eighth, 1546. Reigned 6 Years, 5 Months, 19 Days.

 Years of his Reign. Anno Dom. Years expired, Mar. 25. 1672. 1 1547 125 2 1548 124 3 1549 123 4 1550 122 5 1551 121 6 1552 120

MARY, Began her Reign July the Sixth, 1553. Reigned 5 Years, 4 Months, 22 Days.

 Years of her Reign. Anno Dom. Years expired, Mar. 25. 1672. 1 1554 118 2 1555 117 3 1556 116 4 1557 115 5 1558 114

ELIZABETH, Began her Reign November the Seventeenth, 1558. Reigned 44 Years, 4 Months, 15 Days.

 Years of her Reign. Anno Dom. Years expired, Mar. 25. 1672. 1 1559 113 2 1560 112 3 1561 111 4 1562 110 5 1563 109 6 1564 108 7 1565 107 8 1566 106 9 1567 105 10 1568 104 11 1569 103 12 1570 102 13 1571 101 14 1572 100 15 1573 99 16 1574 98 17 1575 97 18 1576 96 19 1577 95 20 1578 94 21 1579 93 22 1580 92 23 1581 91 24 1582 90 25 1583 89 26 1584 88 27 1585 87 28 1586 86 29 1587 85 30 1588 84 31 1589 83 32 1590 82 [Page 6]33 1591 81 34 1592 80 35 1593 79 36 1594 78 37 1595 77 38 1596 76 39 1597 75 40 1598 74 41 1599 73 42 1600 72 43 1601 71 44 1602 70

JAMES, Began his Reign March the Twenty Fourth, 1602. Reigned 22 Years, 0 Months, 3 Days.

 Years of his Reign. Anno Dom. Years expired, Mar. 25. 1672. 1 1603 69 2 1604 68 3 1605 67 4 1606 66 5 1607 65 6 168 64 7 1609 63 8 1610 62 9 1611 61 10 1612 60 11 1613 59 12 1614 58 13 1615 57 14 1616 56 15 1617 55 [Page 7]16 1618 54 17 1619 53 18 1620 52 19 1621 51 20 1622 50 21 1623 49 22 1624 48

CHARLES I. Began his Reign March the Twenty Seventh, 1625. Reigned 23 Years, 11 Months, 0 Days.

 Years of his Reign. Anno Dom. Years expired, Mar. 25. 1672. 1 1625 47 2 1626 46 3 1627 45 4 1628 44 5 1629 43 6 1630 42 7 1631 41 8 1632 40 9 1633 39 10 1634 38 11 1635 37 12 1636 36 13 1637 35 14 1638 34 15 1639 33 16 1640 32 17 1641 31 18 1642 30 19 1643 29 20 1644 28 [Page 8]21 1645 27 22 1646 26 23 1647 25 24 1648 24

CHARLES II. Began his Reign January the Thirtieth, 1648. and is now Reigning.

 Years of His Reign. Anno Dom. Years expired, Mar. 25. 1672. 1 1649 23 2 1650 22 3 1651 21 4 1652 20 5 1653 19 6 1654 18 7 1655 17 8 1656 16 9 1657 15 10 1658 14 11 1659 13 12 1660 12 13 1661 11 14 1662 10 15 1663 9 16 1664 8 17 1665 7 18 1666 6 19 1667 5 20 1668 4 21 1669 3 22 1670 2 23 1671 1 24 1672 0

### Advice touching the POSTS, and ROADS, more exactly than hath hitherto been published.

I. Concerning Letters, which may be sent from LONDON.

On Mondays, TO France, Spain, Italy, Germany, Flanders and Sweedland, Denmark, Kent and the Downs.

On Tuesdays, To Holland, Germany, Sweedland, Denmark, Ireland, Scotland, all parts of England and Wales.

On Wednesdays, To all parts of Kent and the Downs.

On Thursdays, To France, Spain, Italy, all parts of England and Scotland.

On Fridays, To Flanders, Germany, Italy, Sweedland, Den­mark, Holland, Kent and the Downs.

On Saturdays, All parts of England, Wales, Scotland, Ireland.

Letters are returned from all parts of England and Scotland, certainly every Monday, Wednesday, and Friday; from Wales every Monday and Friday; and from Kent and the Downs every day; But from other parts more uncertainly, in regard of the Sea.

### POST STAGES.

 NORTHERN-ROAD. Miles. Total. FRom London to Waltham-cross 12 12 From Waltham-crose to Ware 8 20 From Ware to Royston 13 33 From Royston to Caxton 8 41 From Caxton to Huntington 9 50 From Huntington to Stilton 9 59 From Stilton to Stamford 12 71 From Stamford to Southwitham 8 79 From Southwitham to Grantham 8 87 From Grantham to Newark 10 97 From Newark to Tuxford 10 107 From Tuxford to Bautry 12 119 From Bautry to Doncaster 6 125 From Doncaster to Ferrybrigg 10 135 From Ferrybrigg to Tadcaster 9 144 From Tadcaster to York 8 152 From York to Burrowbrigg 12 164 From Burrowbrigg to North-allerton 12 176 From North-allerton to Darlington 10 186 From Darlington to Durham 14 200 From Durham to Newcastle 12 212 From Newcastle to Morpeth 12 224 From Morpeth to Alnwik 12 236 From Alnwik to Belford 12 248 From Belford to Berwick 12 260 From Berwick to Cockburnspeth 14 274 From Cockburnspeth to Haddington 14 288 From Haddington to Edenbrough 12 300

 [Page 11]NORWICH-ROAD. Miles. Total. From Royston to Cambridge 10 10 From Cambridge to Newmarket 10 20 From Newmarket to Bury 10 30 From Bury to Thetford 10 40 From Thetford to Attlebrough 10 50 From Attlebrough to Norwich 12 62

 CHESTER-ROAD. Miles. Total. From London to Barnet 10 10 From Barnet to St. Albans 10 20 From St. Albans to Dunstable 10 30 From Dunstable to Fenistratford 8 38 From Fenistratford to Tosseter 12 50 From Tosseter to Daintry 10 60 From Daintry to Coventry 14 74 From Coventry to Coshall 8 82 From Coshall to Lichfield 12 94 From Lichfield to Stone 16 110 From Stone to Namptwich 16 126 From Namptwich to Chester 14 140 From Chester to Northope 18 158 From Northope to Denbeigh 12 170 From Denbeigh to Conway 14 184 From Conway to Blewmorris 10 194 From Blewmorris to Hollyhead 24 218

 DARBY-ROAD. Miles. Total. From Tosseter to Northampton 6 6 From Northampton to Harbrough 12 18 From Harbrough to Leicester 12 30 From Leicester to Loubrough 8 38 From Loubrough to Darby 12 50

 [Page 12]WESTERN-ROAD. Miles. Total. From London to Stanes 16 16 From Stanes to Hartford bridge 16 32 From Hartford bridge to Basingstoke 9 41 From Basingstoke to Andover 18 59 From Andover to Salisbury 16 75 From Salisbury to Shaftsbury 19 94 From Shaftsbury to Sherborn 16 110 From Sherborn to Crookhorn 13 123 From Crookhorn to Huniton 19 142 From Huniton to Exeter 15 157 From Exeter to Ashburton 20 177 From Ashburton to Plymouth 24 201 From Plymouth to Foye From Foye to Trowro From Trowro to Merkejew

 PORTSMOUTH-ROAD. Miles. Total. From Stanes to Hartford bridge 16 16 From Hartford bridge to Petersfield 10 26 From Petersfield to Portsmouth 24 50

 YARMOUTH-ROAD. Miles. Total. From London to Burntwood 16 16 From Burntwood to Witham 18 34 From Witham to Colchester 12 46 From Colchester to Ipswich 16 62 From Ipswich to Saxmundum 16 78 From Saxmundum to Beckles 16 94 From Beckles to Yarmouth 10 104

 [Page 13]BRISTOL-ROAD. Miles. Total. From London to Huntslo 10 10 From Huntslo to Maidenhead 16 26 From Maidenhead to Reading 12 38 From Reading to Newberey 16 54 From Newberey to Malbrough 15 69 From Malbrough to Chippenham 15 84 From Chippenham to Bristol 20 104

 GLOCESTER-ROAD. Miles. Total. From Maidenhead to Abbington 16 16 From Abbington to Farrington 15 31 From Farrington to Cirencister 15 46 From Cirencister to Glocester 18 64

 KENTISH-ROAD. Miles. Total. From London to Dartford 14 14 From Dartford to Rochester 14 28 From Rochester to Sittingburn 12 40 From Sittingburn to Canterbury 15 55 From Canterbury to Deal Dover 15 7 [...]

### Forreign Weights and Measures, carefully compared with the English, by the great pains and industry of the famous, and my worthy Friend, Sir Jonas Moor, Knight.

 English Foot, into 1000 e­qual parts. English foot, in­to inches, and tenth parts of an inch. The pound A­verdupois into 100 parts. London Foot 1000 0.12.0 100 France. Paris, the Royal Foot 1.068 1.00.8 0.93 Lyon Ell 3.976 3.11.7 1.09 Boloyne Ell 2.076 2.00.8 0.89 The 17 Provinces. Amsterdam Foot 0 942 0.11.3 0.93 Ell 2.269 2.03.2 Antwerp Foot .946 0.11.3 0.98 Ell 2.273 2.03.3 Brill Foot 1.103 1.01.2 Dort Foot 1.184 1.02.2 Rynland or Leyden foot 1.033 1.00.4 0.96 Ell 2.260 2.03.1 Lorain Foot .958 0.11.4 0.98 Mecalin Foot .919 0.11.0 0.98 Middlebourg Foot .991 0.11.9 0.98 [Page 15] Germany. Strasbourgh Foot .920 0.11.0 0.93 Bremen Foot .964 0.11.6 0.94 Cologn Foot .954 0.11.4 0.97 Frankford and Me­nain Foot .948 .11.4 0.93 Ell 1.826 1.09.9 Hambrough Ell 1.905 1.10.8 0.95 Leipsig Ell 2.260 2.03.1 1.17 Lubick Ell 1.903 1.09.8 Noremburgh 1.006 1.00.1 0.94 Ell 2.227 2.03.3 Bavaria .954 0.11.4 Vienna 1.053 1.00.6 0.83 Spain and Portugal. Spanish Palm, or the Palm of Castile. .751 0.09.0 0.99 The Spanish Vare or Rod, (four Palms) 3.004 3.00.0 Their Foot is ⅓ of the Vare 1.001 1.00.0 Lisbon Vare 2.750 2.09.0 1.06 Gibralter Vare 2.760 2.09.1 1.03 Toledo Foot .899 0.10.7 1.00 Vare 2.685 2.08.2 Italy. Roman Foot, on the Monum. of Cossutius .967 0.11.6 1.23 Of Statelius .972 0.11.7 Roman Palm, for building, where­of ten make the Cauna .732 0.08.8 [Page 16] Bononia Foot 1.204 1.02.4 1.27 Ell 2.147 2.01.7 Perch, whereof 500 to a Mile. 12.040 12.00.5 Florence Brace or Ell 1.913 1.11.0 1.23 Naples Palm .861 0.09.6 1.43 Brace 2.100 2.01.2 Cauna 6.880 6.10.5 Genua Palm .830 .09.6 1.42 Mantoua Foot 1.569 1.06.8 1.43 Milan Calamus 6.544 6.06.5 1.40 Parma Cubit 1.866 1.10.4 1.43 Venice Foot 1.162 1.01.9 1.53 Other Places. Dantzick Foot .944 0.11.3 1.19 Ell 1.903 1.10.8 Copenhagen Foot .965 .11.6 0.94 Prague (in Bohemia) Foot 1.026 1.00.3 1.06 Riga Foot 1.831 1.09.9 China Cubit 1.016 1.00.2 Turin Foot 1.062 1.00.7 Cairo Cubit 1.824 1.09.9 1.61 Persian Arash 3.197 3.02.3 Turkish Pike, at Con­stantinop. the greater 2.200 2.02.4 0.86 The Greek Foot 1.007 1.00.1 Montons Universal foot 0.675 .08.11 A Pendulum of the just length whereof will Vi­brate 132 times in a Minute.
Ex. by me, Jonas Moore,

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