A Method for Finding the Number of the Julian Period for Any Year Assign'd, the Number of the Cycle of the Sun, the Cycle of the Moon, and of the Indictions for the Same Year,being Given; Together with the Demonstration of That Method

A Method For finding the Number of the Julian Period for any Year assign'd, the Number of the Cycle of the Sun, the Cycle of the Moon, and of the Indictions for the same year being given; together with the Demonstration of that Method. In these Transactions, No. 18, p. 324, is a Theorem for finding the Year of the Julian Period by a new and very easy Method, which was taken out of the Journal des Savans, No. 36, as it had been proposed and communicated by the Learned Jesuit De Billy. Multiply the \[ \begin{array}{c} \text{Solar Cycle} \\ \text{Lunar Cycle} \\ \text{Indication} \end{array} \] by \[ \begin{array}{c} 4845 \\ 4200 \\ 6916 \end{array} \] Then divide the sum of the Products by 7980 (the Julian Period) the Remainder of the Division, without having regard to the Quotient, shall be the Year inquired after. Some Learned Mathematicians of Paris, to whom the said P. de Billy did propose this Problem, have found the Demonstration thereof as the same Journal intimates. There being no further Elucidation of the said Theorem since publish'd, Mr. John Collins, now a Member of the Royal Society, communicated what follows, viz. That the Julian Period is a Basis, whereon to found Chronology not liable to Controversie, as the Age of the World is: And 'tis the Number above said, to wit, 7980, which is the Product of \[ \begin{array}{c} 28 \\ 19 \\ 155 \end{array} \] the \[ \begin{array}{c} \text{Solar Cycle} \\ \text{Lunar Cycle} \\ \text{Indication} \end{array} \] Concerning this Julian Period, the late Archbishop of Armagh, Usher, in the Preface to his Learned Annals, advertiseth, that Robert Lotharing, Bishop of Hereford, first observed the Conveniencies thereof; 500 years after whom, it was fitted for Chronological uses by Joseph Scaliger, and is now embraced by the Learned as such a limit to Chronology, that within the space of 7980 Years, the Number of the Sun's Cycle, the Prime, and the year of the Roman Indiction (which relates to their ancient Laws and and Records) can never happen alike. And these Remarques being given, the year of the Julian Period is by the former Rule infallibly found. This Period is used by the said Archbishop in his Annals, and is by him accounted to exceed the Age of the World 709 years. Those that desire further satisfaction about Era's, Epoch's, and Periods, may repair to many Authors, and among them to Gregory's Posthuma in English, Helvici Chronologia, Aegidii Strauchii Breviarium Chronologicum, who is one of the latest Authors. Now as to the Problem itself, it may be thus proposed: Any number of Divisors, together with their Remainders after Division, being proposed, to find the Dividend. This thus generally proposed is no new Problem, and was resolved long since, by John Geysius, by the help of particular Multipliers, such as those above-mentioned, and publish'd by Alstedius in his Encyclopaedia in Ann. 1630. and by Van Schooten in his Miscellanies. We shall clear up, what Authors have omitted concerning the Definition and Demonstration of such fixed Multipliers, &c. And therefore say, that each Multiplier is relative to the Divisor, to which it belongs, and thus define it: It is such a Number, as divided by the rest of the Divisors, or their Product, the Remainder is 0; but divided by its own Divisor, the Remainder is an Unit. We require the Divisors proposed to be Primitive each to other, i.e. that no two or more of them can be reduced to lesser terms by any common Divisor: For if so, the Question may be possible in itself, but not resolvable by help of such Multipliers, such being impossible to be found. The reason is, because the Product of an Odd and Even Number is always Even, and that divided by an Even Number, leaves either Nothing, or an Even Number. \[ \begin{align*} \text{Divisors} & \quad \text{The Multipliers relative thereto are} \\ 28 & \quad 4845 \\ 19 & \quad 4200 \\ 15 & \quad 6916 \end{align*} \] The Definition affords light enough for the discovery of these Numbers. To instance in the first: The Product of 19 and 15 is is 285, which multiply by all numbers successively, and divide by 28, till you find the Remainder required. Thus twice 285 is 570, which divided by 28, the remainder is 10; also thrice 285 is 855, which divided by 28, the remainder is 15. Thus if you try on successively, you'll find, that 17 times 285, which is 4845, is the Number required, the which divided by 28, the Remainder is an Unit. Hence then we shall find, that \[ \begin{align*} 4845 & \text{ is equal to the Solid or Product of } \\ 4200 & \text{ } 19, 15, 17. \\ 6916 & \text{ } 23, 15, 10. \\ & \text{ } 28, 19, 13. \end{align*} \] More easy ways of performing this postulatum, are to be found in Van Schooten's Miscellanies, and Tacquet's Arithmetick, which perchance are not so obvious to every understanding. For illustration of the Rule proposed, take this Example. | In the year | Cyclo Solis | Cyclo Luna | Indictio | |-------------|-------------|------------|----------| | 1668 | 25 | 16 | 6 | The Sum of the Products 229821 The which divided by 7980, the remainder is 6381, for the year of the Julian Period; from which subtracting 709, there remains 5672, for the Age of the World, according to Archbishop Usher. For DEMONSTRATION of this Rule we thus argue: 1. Each Multiplier multiplied by its Remainder, is measured or divided by its own Divisor, leaving such a Remainder as is proposed. For before, each Multiplier was defined to be a Multiplex of its own Divisor, plus an Unit: Wherefore multiplying it by any Remainder, it doth only render it a greater Multiplex in the said Divisor, plus an Unit, multiplied by the Remainder, which is no other than the Remainder itself; but if o remain, that Product is destroyed. 2. The Sum of the Products, divided by each respective Divisor, have the Remainder assigned. For concerning the first Product, it is by the first Section measur'd sur'd by its own Divisor, leaving the remainder proposed; and if we add the rest of the Products thereto, we only add a Multiplex of its own Divisor, which in Division enlargeth the Quote, but not the Remainder. Particularly the second Multiplier is $28 + 15 + 10 + \text{Remainder}$, all which is but a Multiplex of $28$. And so the third Product is $28 + 19 + 13 + \text{Remainder}$. And what hath been said concerning the sum of the Products, being divided by the first Divisor, and leaving the Remainder thereto assign'd, may be said of each respectively. 3 The sum of the Products divided by the solid of the three Divisors, leaves a Remainder so qualified as the said Sum. For concerning the said Sum, 'tis evident by the second hereof, that it is no other than the first Product, increas'd by adding a just Multiplex of the first Divisor, that thereby we did only enlarge the Quote, not alter the Remainder. By the like reason, the subtracting a just Multiplex thereof, doth only alter the Quote, not the Remainder; but the Solid of all three Divisors, multiplied here by the Quote, as there by the Remainder, is no other than a just Multiplex of the first Divisor. Wherefore the Remainder, after this Division is perform'd, is of the same Quality as the sum of the Products, and divided by the first Divisor, leaves the Remainder proper thereto. And the like may be said concerning each Divisor. As in the Method hitherto deliver'd, we requir'd the Divisors be Primitive to each other; so, if we take the Problem as generally proposed, in the Preface to Helvius his Chronologia, we are told, common Arithmetick fails in the solution thereof, and Tacquet denies it to be performable by the Regula Falsi, and being unlimited, we must do it by Tryals. Wherefore, When any two Divisors with their Remainders are proposed, try the Multiplices of one of them, increased by its Remainder, and divide by the other: If you find such Remainders as are not for the purpose, and that they are repeated, the Problem is impossible. Example. Divisors Remainders 6. 3. 8. 5. The Multiplices of 8, increased by 5, are 13, 21, 29, 37, 45, 53. Those divided by 6, the Remainders are 1, 3, 5, 1, 3, 5. Here you see 21 and 45 for the purpose, and take the Progression, adding the common Difference 24 (which is the least Dividend measured by 6 and 8) and you have 21, 45, 69, 93, 117, 141. Admit, the Question had concerned these three Divisors: Then dividing the former Progression by 9, the Remainders are 3, 0, 6, 3, 0, 6. Wherefore I conclude, that the third and sixth of these Numbers are those sought, to wit, 69 or 141, and so on progressively; whereas, if you had propounded the Remainder of 9 to have been any other Number than 3, 0, 6, the Problem, as concerning all these, had not been possible. Some easy Cases of the Problem are these: When the Remainder of some Divisor is 0, and of each of the rest of the Divisors, an Unit, or less by an Unit, than the Divisor. In which Cases you are to find such a Multiplex of the Product or least Dividend measurable by those Divisors that have Remainders, which increased or diminished by an Unit, may be a just Multiplex of that Divisor that hath no Remainder. These Cases are handled by Tacquet, and Bachet in his Problemes plaisans & delectables. PROBLEM. To find the Year of the Julian Period for any year of our Lord proposed. It is necessary to be furnished with the Sun's Cycle, the Prime Number, and the Number of the Roman Indiction, which the industrious Mr. Street thus performs: When When 1. 9. 3. to the Year hath added been, Divide by 19. 28, fifteen. The Remainders are the Numbers sought. And hereby we found them for the year 1668, in the former Example The use of the Prime is, to find the Epact, and thereby the Moons Age, time of High Water, &c. A farther use of the Suns Cycle is, to attain the Dominical Letter, and thereby to know the Day of the Week, on which any Day of the Month happens. But this is more easily and with less caution obtain'd, by finding on what Day of the Week the first of March happens for ever, according to such Rules and Verses as I have elsewhere published. In brief thus: To the Number . . . . . . . . . . 2. Add the Year of our Lord, suppose 1669. And its even 4th part, neglecting ? what remains, if any . . . . . . . . . . 417. The Sum . . . . . . . . . . 2088. Divide by 7, noting the Remainder, which shews the Number of the Day of the Week, accounting Sunday first. If o remain, the first of March falls on a Saturday. In this Example there remains 2, shewing the first of March to fall on Monday. If it were required to perform this for years preceding our Saviour's Nativity, then take this Rule: To the Year add its even fourth part, the Sum divide by 7, the Remainder shews the Day of the Week, accounting Sunday first, Saturday second, and so backward. PROBLEM. To find what day of the Month in the first week of each Month, happens to be on the same day of the Week as the first of March. Use the (plain) following Verses, in which the twelve Words relate to the twelve Months of the Year, accounting March the first: Ask endless Comfort, God enough bestows, From Divine Axioms Faith confirmed grows. The Alphabetical Number of the first Letter of the word, proper to the Month proposed, is the Answer. Example. If the Month were April, the word proper thereto is Endless, and E is the fifth Letter in the Alphabet. Wherefore conclude, That the first of March and fifth of April do forever happen on the same day of the Week; which for the year 1669, will be on Monday. PROBLEM. To find on what day of the Week the first day of each Month happeneth. Supposing the first of March known, it might be reckoned from the former Problem, but the following Verse, beginning with March, as the former, is more ready for the purpose: A dreadful Fire, Beholders daily Gaze, Chastized England. Ah cruel fatal Blaze. Explication. In the Year 1669, the first of March is Monday; I would know on what day of the week the first of October happens. The word proper to the Month is England; then count Alphabetically to E, viz. A. Monday, B. Tuesday, C. Wednesday, D. Thursday, E. Friday, which is the day sought. Whence conclude, that the 1st, 8th, 15th, 22th, 29th days of October are all Fridays. Thence it is easy to reckon, on what day of the Week any day of that Month happened; and so for all other Months. PROBLEM. To find on what Day of the Month the Sun enters into any Sign of the Zodiac. For this, ex super abundanti, we give the following Verse: Charles brought Content, divers Effects ensue, Envy, Fear, Dolors, Danger, kids adieu. Here again the twelve Words relate to the twelve Months; March being the first. To To the number of the Letter of the Alphabet the word begins with, add 7. Example. Bear is the word for October, and F the sixth Letter: Wherefore the Sun enters into the 8th Sign, to wit, Scorpio, on the 13th of October. An Account of some Books. I. PETRI LAMBECHII LIB. PRIMUS PRODROMI HISTORIÆ LITERARIIÆ, &c. The Author of this Book is now the Historiographer and Library-keeper to the Emperour. He publish'd this Volume some few years ago at Hamburgh, the place of his Birth, (whence an Exemplar was but lately sent to the Publisher.) He was excited to this Work by the complaint made by the illustrious Lord Verulam, (Lib. 2. cap. 4. de Augm. Scientiarum) of the want of a compleat History of Learning, that might give a satisfactory Account of the Rise, Progress, Transmigrations, Interruptions, Declinations, and Restaurations of all kind of Learning, Sciences, Arts, and Inventions; together with the occasion of Inventions through all Arts; the method of teaching, and the manner of improving and advancing them: Adding the various Sects, and the most famous Controversies among the Learned; the Encouragements they received; the chief Writings they composed; their Schools, Academies, Societies, Colleges, Successions, Orders, and whatever belongs to the state of Learning. This grand Desideratum our Author undertakes to supply the World with, and in order thereunto, hath given us the first Book of the Prodromus of this History, and with it the four first Chapters of the Second Book, together with an Appendix, containing a Summary of the chief Persons and Things he intends more fully and accurately to treat of in the remaining 32 Chapters, designed for the same second Book: To which, he subjoins two Tables of Universal Chronography, in the first whereof he exhibits the succession of all Ages from the Creation of the World to the beginning of the common Christian Account; in the other, a Continuation of them from the beginning of the said Account unto this present Age: In which Tables he gives a general Idea of the Connexion of all Ages, as they are computed in respect of the Vulgar