Category Johnson’s Universal Cyclopaedia – 1887

Gauss

Bran

I am taking a class on linear algebra at the moment, which means that I am breathing in matrices and Gaussian elimination on a daily basis. There is a footnote in our textbook where row reduction algorithms are discussed: "The algorithm here is a variant of what is commonly called Gaussian elimination. A similar elimination method for linear systems was used by Chinese mathematicians in about 250 B.C. The process was unknown in Western culture until the nineteenth century, when a famous German mathematician, Carl Friedrich Gauss, discovered it. A German engineer, Wilhelm Jordan, popularized the algorithm in an 1888 text on geodesy" (Linear Algebra and its Applications, 4th edition, by David C. Lay - 2012 - p. 12). Geodesy is mathematically determining the size and shape of the Earth. I remembered Gauss's name coming up in my statistics class; it turns out his contributions to both linear algebra and statistics were secondary byproducts of his interest in geodesy. I decided to look him up in my reference works and discovered that his contributions to mathematics and science, in general, were so great that Gaussian elimination isn't even mentioned.

Even though it was just published 32 years after Gauss's death, Johnson's Universal Cyclopaedia (1887) features a compact summary of his life and accomplishments: Gauss (Karl Friedrich), b. in Brunswick, Germany, Apr. 30, 1777; was educated at the expense of the duke of Brunswick, who had heard of his precocious mathematical talents; solved when eighteen years old the problem of the division of the circle into seventeen equal parts, and afterwards became famous for skill in the indeterminate analysis and in curious numerical questions; demonstrated Fermat's theorem; became in 1807 professor of astronomy at Göttingen and director of the observatory; received in 1810 the Lalande medal for calculating by a new method the orbits of Ceres and Pallas; was made in 1816 a court councillor, and in 1845 a privy councillor of Hanover; made after 1821 important improvements in geodetic methods and instruments; after 1831 devoted much attention to terrestrial magnetism. D. at Göttingen Feb 23, 1855. Gauss is regarded as one of the first mathematicians of this century (vol. 3, p. 403).

The 11th edition Encyclopaedia Britannica (1910) is a bit more thorough. The article on Gauss can be found in volume 11, beginning with the fact that he was "born of humble parents" (535). Some fun extracts (pp. 535-536):

In 1807 he was appointed director of the Göttingen observatory, an office which he retained to his death: it is said that he never slept away from under the roof of his observatory, except on one occasion, when he accepted an invitation from Baron von Humboldt to attend a meeting of natural philosophers in Berlin. [...] With [Wilhelm] Weber's assistance he erected in 1833 in Göttingen a magnetic observatory free from iron (as Humboldt and F. J. D. Arago had previously done on a smaller scale), where he made magnetic observations, and from this same observatory he sent telegraphic signals to the neighboring town, thus showing the practicality of an electromagnetic telegraph. [...] Running through these volumes in order, we have in the second the memoir, Summatio quarundam serierum singularium, the memoirs on the theory of biquadratic residues, in which the notion of complex numbers of the form a + bi was first introduced into the theory of numbers; and included in the Nachlass are some valuable tables. That for the conversion of a fraction into decimals (giving the complete period for all the prime numbers up to 997) is a specimen of the extraordinary love which Gauss had for long arithmetical calculations, and the amount of work gone through in the construction of the table of the number of the classes of binary quadratic forms must also have been tremendous.

A much longer entry on Gauss appears in the Macropaedia section of the 15th-edition Encyclopaedia Britannica (here the 1997 printing, volume 19, pp. 697-698). This article comments a little bit on his personal life outside of his mathematical discoveries, and also mentions his contributions to statistics (couched in the context of his interest in geodesy); here is a fraction of the details found within:

His own dictum, "Mathematics, queen of the sciences, and arithmetic, the queen of mathematics," aptly conveys his perception of the pivotal role of mathematics in science. [...] His first wife died in 1809, after a marriage of four years and soon after the birth of their third child. From his second marriage (1810-31) were born two sons and a daughter. [...] By introducing what is now known as the Gaussian error curve, he showed how probability could be represented by a bell-shaped curve, commonly called the normal curve of variation, which is basic to descriptions of statistically distributed data. [...] The most important result of their [Weber + Gauss's] work in electromagnetism was the development, by other workers, of electric telegraphy. Because their finances were limited, their experiments were on a small scale; Gauss was rather frightened at the thought of worldwide communication. [...] Teaching was his only aversion, and, thus, he had only a few students. Instead he effected the development of mathematics through his publications, about 155 titles, to which he devoted the greatest care. Three principles guided his work: "Pauca, sed matura" ("Few, but ripe"), his favourite saying; the motto "Ut nihil amplius desiderandum relictum sit" ("That nothing further remains to be done"); and his requirement of utmost rigour. It is evident from his posthumous works that there are extensive and important papers that he never published because, in his opinion, they did not satisfy one of these principles. He pursued a research topic in mathematics only when he might anticipate meaningful relationships of ideas and results that were commendable because of their elegance or generality.

A fun text in general for looking at the lives of great scientists and mathematicians comes from another prolific writer, Isaac Asimov, in Asimov's Biographical Encyclopedia of Science and Technology: The Lives & Achievements of 1195 Great Scientists from Ancient Times to the Present; I have the revised version from 1972. The entries in this book are arranged chronologically, but there is a handy alphabetical index at the front of the book, which helped me quickly locate the biography of Gauss on pages 249-251. My favorite quips:

At the age of three, he was already correcting his father's sums, and all his life he kept all sorts of numerical records, even useless ones such as the length of the lives of famous men, in days. He was virtually mad over numbers. [...] All of this was not without a price, for his intense concentration on the great work that poured form him withdrew him sometimes from contact with humanity. There is a story that when he was told, in 1807, that his wife was dying, he looked up from the problem that engaged him and muttered, "Tell her to wait a moment till I'm through." [...] His agile mind never seemed to cease. At the age of sixty-two he taught himself Russian. [...] Each of his two wives died young and only one of his six children survived him. His life was filled with personal tragedy, and though he died wealthy, he also died embittered.

Knitting

Bran

I just upgraded my knitting needles. I have been using KnitPro / Knitter's Pride / KnitPicks (they're all the same!) interchangeable needles since February 2012, when I purchased the KnitPro Nova Deluxe Set for 43.85€. I was a graduate student in Germany, and that was an insanely expensive purchase for me at the time. They've served me well - they are essentially hollow brass needles coated with nickel plating that screw on to flexible cords of different lengths. I slowly supplemented the initial set over the years with additional needle and cord sizes, but there have been a few issues. Quality control has sometimes been lacking - one pair of needle tips completely lacked the screw connection to attach to the cable, for example, and the actual needle sizes have not always matched up to the claimed size. Over the past couple of years all of them, whether I've actively used them or kept them secure in their case, have developed some sort of sticky film that makes knitting difficult. So, after consulting the internet and a dear friend, I decided to splurge on a set of ChiaoGoo stainless steel interchangeable needles, which arrived this last week. So far I'm enjoying them - no goo and the cables are much more flexible than my other set.

Knitting doesn't get much attention in early encyclopedia sets - if there is an entry on the subject, it tends to be short. Here's the entry from Johnson's Universal Cyclopædia (1887), found in volume 4 on page 613:

Knit'ting [Ang. Sax. cnyttan or knittan; Ger. knutten, knot; Hind. ganth; Sans. gnanthi, a "knot"], a manner of weaving or twisting a single thread into a kind of cloth by means of steel, ivory, or wooden implements called knitting-needles, which are made of various sizes, according to the fineness of thread used and the tightness of stitch required.

Knitting is a far more modern invention than its kindred art, netting. Many antiquaries affirm that knitting was invented in Scotland, and thence introduced into France; others say that it is of Spanish origin, and was first known in England in the reign of Henry VIII. But in a rare collection of the acts of Edward VI. is one specifying, among other woollen articles, "knitte hose, knitte peticotes, knitte gloves, knitte slieves." In 1527 the French knitters formed themselves into a corporation, styled "Communauté des Maîtres Bonnetiers au Tricot," choosing for their patron St. Fiacre.

The 11th edition (1911) of the Encyclopædia Britannica's entry appears in volume 15, p. 869:

KNITTING (from O.E. cnyttan, to knit; cf. Ger. Knütten; the root is seen in "knot"), the art of forming a single thread or strand of yarn into a texture or fabric of a loop structure, by employing needles or wires. "Crochet" work is an analogous art in its simplest form. It consists of forming a single thread into a single chain of loops. All warp knit fabrics are built on this structure. Knitting may be said to be divided into two principles, viz. (1) hand knitting and (2) frame-work knitting (see Hosiery). In hand knitting, the wires, pins or needles used are of different lengths or gauges, according to the class of work wanted to be produced. They are made of steel, bone, wood or ivory. Some are headed to prevent the loops from slipping over the ends. Flat or selvedged work can only be produced on them. Others are pointed at both ends, and by employing three or more a circular or circular-shaped fabric can be made. In hand knitting each loop is formed and thrown off individually and in rotation and is left hanging on the new loop formed. The cotton, wool and silk fibres are the principle materials from which knitting yarns are manufactured, wool being the most important and most largely used. "Lamb's-wool," "wheeling," "fingering" and worsted yarns are all produced from the wool fibre, but may differ in size or fineness and quality. Those yarns are largely used in the production of knitted underwear. Hand knitting is to-day principally practised as a domestic art, but in some of the remote parts of Scotland and Ireland it is prosecuted as an industry to some extent. In the Shetland Islands the wool of the native sheep is spun, and used in its natural colour, being manufactured into shawls, scarfs, ladies' jackets, &c. The principal trade of other districts is hose and half-hose, made from the wool of the sheep native to the district. The formation of the stitches in knitting may be varied in a great many ways, by "purling" (knitting or throwing loops to back and front in rib form), "slipping" loops, taking up and casting off and working in various coloured yarns to form stripes, patterns, &c. The articles may be shaped according to the manner in which the wires and yarns are manipulated.

Here is the Encyclopedia Americana (1924) on the subject (vol.16, pages 488):

KNITTING, an industrial and ornamental art akin to weaving, but of much later origin. It does not appear to be more than three or four centuries old, and seems to have been first used in the manufacture of stockings. It consists in forming a series of loops with a single thread, through which another row of loops is passed, and so on consecutively in spiraled circles, the garment being shaped by variations in the number of loops in a row. In hand-knitting, steel-wires or bone or composition needles are used, termed knitting needles, and on these the loops are formed. For manufacturing purposes hand-knitting has been entirely superseded by machinery (see Knitting-Machine), which is constantly receiving new improvements. Hand-knitting, however, still forms an agreeable domestic occupation and also furnishes many women in some parts of the world with means of subsistence. Promptly upon America's entrance into the World War, many patriotic societies, and women of the Red Cross in particular, started a work of knitting sweaters for soldiers and sailors. The movement was taken up by thousands of women, who devoted their otherwise idle time to knitting, and it became common to see women everywhere with knitting bags on their arms, that they might work whenever they had spare moments. On the street cars, at social gatherings, in intervals of business, there was industrious knitting, resulting in a great volume of very serviceable sweaters and some other knitted garments being provided for the "boys at the front."

By mid-century, knitting started getting more attention in the encyclopedia sets. Hand-knitting didn't appear at all in my 1919 World Book, for example, but my 1958 set features a diagram on how to "cast on" stitches and notes that "knitting is older than written history." The article (volume 10, pages 4188-4189) also comments on knitting education: Knitting is taught to girls in some European schools. Schools in the United States do not usually teach knitting. The reason is that there are only two main stitches used in knitting — the knit and the purl — and the rest depends upon practice alone. Department stores and small "knit shops" in the United States have made knitting a popular hobby. Hand knitting reached a peak in the United States in 1935, partly because stores selling yarn offered free lessons in knitting. I wonder what the mid-century World Book would think of "stitch-and-bitch" sessions.

The 14th edition of the Encyclopædia Britannica (my copy is from 1965) concentrates on machine knitting under "Knitting," but directs the reader interested in hand-knitting to the article on "Needlework" (vol. 16, p. 183), which describes with diagrams (see picture below) the basics of the craft. I, for one, would not want these illustrations to be my sole reference for learning the craft, but it shows the trend towards including illustration. In general, the newer the encyclopedia set, the more likely there will be both illustrative examples of what knitted work looks like and diagrams demonstrating the basic stitches. The 1992 World Book, for example, features two-color diagrams illustrating the stitches as well as full-color close-up photographs of finished stockinette and garter-stitch knitting.

Illustration from the 14th Edition Encyclopædia Britannica, "Needlework"

The Calculation of Easter

Bran

I find it very amusing that the entirety of the very lengthy article on "Easter" found in Johnson's Universal Cyclopædia (1887) is devoted to calculating when the holiday actually takes place. It includes a copy of an instrument called "The Churchman's Companion to the Calendar," with instructions on how to use this system of rings to calculate the date for a given year. I'll attempt to calculate Easter for the current year following the article (bottom of this post). This entry is from volume 2, pages 558 - 561:

Eas'ter [Ger. Oster; Gr. πάσχα; Lat. pas'cha; Fr. pâques; etymology doubtful], the principal festival of the Christian year, observed in commemoration of the resurrection of our blessed Lord. The returns of this anniversary were originally regulated, and in imitation of this early usage have always continued to be, by the calendar of Judea, in which the months were conterminous with the revolutions of the moon. A mean lunation being, roughly, twenty-nine and a half days long, twelve lunar months, or a lunar year, fall short of a solar year by about eleven days. The beginning of the Jewish year therefore goes backward on the natural year eleven days annually, requiring an intercalary month to be introduced in the third year, and again in the sixth, ninth, eleventh, fourteenth, and so on. Any anniversary regulated by such a calendar as this is consequently movable in reference to a calendar regulated by the sun. The Resurrection took place just after the Jewish feast of the Passover, which was held on the fourteenth day of Nisan, the first month of the year — that is to say, the fourteenth day of the moon, or not far from the time of the full moon. The Christians of Jerusalem, and after them those of the Asiatic churches generally, were accustomed to hold the feast of Easter on this same day or simultaneously with the Jewish Passover. This usage was unacceptable to the Gentile churches in Italy and the West generally, which preferred to celebrate Easter on the Sunday following the fourteenth day of the moon; and the difference of practice in this particular led to grave dissensions between the East and West, which were at length pacified by the agreement reached in the Council of Nicæa (A. D. 325), to make the Western usage universal. Since this early period Easter has always been observed throughout the world on the Sunday following the fourteenth of that lunation of which this fourteenth day falls on the 21st of March or next later. In order to find the time of Easter for any given year, it would seem that we should calculate the exact time of the new moon in that year for March, and try whether the fourteenth day of that moon (the day of new moon itself being counted the first) would all not earlier than the 21st; in which case the Sunday following this fourteenth day might be presumed to be Easter. But should this fourteenth day fall earlier than the 21st of March, we should conclude that the new moon of April must be taken. The ecclesiastical calendar, however, is only nominally dependent on the moon in the heavens, the true moon and the calendar moon sometimes differing in their age more than two days. The practical reason for this is, that if the astronomical time of new moon is taken, this time will not be the same in the local times of different longitudes; so that a meridian may always be assigned such that the same new moon may fall on different calendar days on different sides of it. And if the calculation is very nicely made, when new moon happens exactly at midnight of Saturday or Sunday in the middle of a large city like London, the east and west halves of the city may have their Easter upon two very different days. The ecclesiastical moon is therefore an ideal or artificial moon; and in determining the beginning and end of each lunation no account is taken of any differences smaller than a day. In order to divest the ecclesiastical calendar as much as possible of complexity, advantage is taken of the fact discovered by Meton, an Athenian astronomer in the fifth century before our own era, that in a period of nineteen solar years the sun and the moon return almost exactly to the same relative positions which they occupied at the beginning of this period, the difference amounting to little more than the space the moon would move over in two hours. The calendar therefore assumes that the moons determining Easter will recur in the same order every nineteen years throughout an entire century, and sometimes throughout two or three centuries. The Easters themselves do not therefore necessarily recur on the same days of the month of March or April in each of these successive series of nineteen years, but would do so if the same days of the week always corresponded to the same days of the month. This, however, is not necessarily the case; and as Easter must be a Sunday, it is necessary, in order to fix definitely the date of Easter in any given year, to know both the place of the year in the series of nineteen (or in the Metonic cycle) and also the day of the week on which the year began, or (what is practically the same thing) the dominical letter for the year. Various methods have been given for finding Easter, but all of them commence, expressly or implicitly, with the determination of these two elements. The rules given by Prof. de Morgan in the "Companion to the British Almanac" for 1845 occupy about a page. The formulæ of Delambre, in the first volume of his "History of Modern Astronomy," and those of Gauss, given in the first volume of the "Theoretical and Practical Astronomy" of the same writer, though concise as mathematical expressions, involve much laborious computation in their practical application. The following rules, however, originally devised by the writer of the present article, are very simple and easy, It is to be observed, first, that the fourteenth day of the Easter moon, being approximately the time of full moon, is called the paschal full moon. The number of the year in the lunar cycle is also called the Golden Number. (See Golden Number.) Then, supposing that we know the golden number and the dominical letter, we find, for the present century, the paschal full moon as follows:

If the golden number is odd: To four times the golden number add ten; and

If the golden number is even: To four times the golden number add twenty-five.

The result, in either case, if greater than twenty and less than fifty, is the date of paschal full moon, considered as a day of March (that is to say, if it happens to be, say, thirty-three, it is the thirty-third of March = the second of April, and so on). If not grater than twenty and less than fifty, add thirty, or subtract thirty, or twice thirty, if necessary to make it so, and the result is once more paschal full moon.

Then, to find Easter: To the constant number eighteen add the numerical value of the dominical letter (i.e. A = 1; B = 2; C = 3, etc.), and the sum, if greater than the value of paschal full moon just found, is the date of Easter; but if not, add seven, or twice seven, or three times seven, and so on till a total is obtained which exceeds that value; and this total is the date of Easter considered as a day of March.

To find the golden number and the dominical letter: In either case first separate the hundreds in the number expressing the given year of our Lord from the years less than a hundred, and treat the parts independently of each other. First, for the dominical letter: If the hundreds be dived by four, the remainder from the division will have one or other of the following values — viz., 0, 1, 2, 3. And the dominical letters belonging to the hundreds which give these remainders respectively will be A, C, E, G = 1, 3, 5, 7. These, for convenience, call centurials. Then for the years take half the largest number divisible by four — i.e. half the number of the latest leap-year — increase this by seven, and subtract the excess of fours (i.e. the remainder left in the previous division of four). To this result add the centurial, and the excess of sevens in the sum will e the value of the dominical letter; it being observed that if there is no excess the dominical letter has the value of seven itself, or is G. Leap-years have two dominical letters — one for January and February; the other, which is less than the former by a unit, for the remainder of the year. This last, which only is used in finding Easter, is that given by the rule.

To find the dominical letter for Old Style the process is the same except as to the centurial. The centurial for old style is found by adding three to the number of hundreds, and suppressing sevens. Thus, if the hundreds be fifteen, we have 15 + 3 = 18. And 18 with seven dropped as often as possible leaves 4, which is the old style centurial. If there is no excess of sevens, the centurial is seven itself.

Secondly, for the golden number: Add a unit to the number expressive of the given year; then divide the years by twenty, and add the quotient to the remainder. Next divide the centuries by four, and add the quotient to five times the remainder. Finally, add the two results, and the sum, if nineteen or less, is the golden number. If it exceeds nineteen, drop nineteen, or, if necessary, twice nineteen, and the number left, being not greater than nineteen, will be the golden number.

Take, as an example, the year 1873. For the dominical letter: 18 ÷ 4 gives 2 remainder, and the centurial is accordingly 5. The number of the largest leap-year in 73 is 72, and the half of this is 36. Then 36 + 7 = 43, and 43 - 1 = 42. Finally, 42 + 5, with the sevens suppressed, is evidently 5 = E, which is the dominical letter of 1873.

For the golden number: 1873 + 1 = 1874. Then, 74 ÷ 20 = 3, with 14 remainder, and 14 + 3 = 17. Also, 18 ÷ 4 = 4, with 2 remainder, and 2 × 5 + 4 = 14. Then, 17 + 14 = 31, and 31 - 19 = 12, the golden number for 1873.

For Easter in 1873: 12 × 4 + 25 = 73. Then 73 - 30 = 43, or paschal full moon is the 43d day of March. To 18 add 5, the value of the dominical letter, and the result, 23, is smaller than the date of the paschal full moon. But 23 + 7 + 7 + 7 = 44, which is greater than that date (43), and Easter is the 44th day of March, or the 13th day of April.

There is one case not provided for in the foregoing. If in finding the paschal full moon we obtain a result which is exactly twenty or exactly fifty, adding or subtracting thirty will not bring it between those limits. In this case paschal full moon must be taken at 49. There is also an irregularity arbitrarily introduced by the mathematicians of Pope Gregory XIII., by whom the calendar was regulated, which is this: Should the rules above laid down give forty-nine directly as the date of paschal full moon, this must be reduced to forty-eight in case the golden number is 12 or upward; not otherwise.

For centuries earlier or later than the present, the rules are the same, except that the numerical terms ten and twenty-five used in finding paschal full moon are liable to variation (but do not always vary) in passing from century to century. The second of these terms always exceeds the first by fifteen. The first may be found for any century up to the forty-second by the following rule: from the number of the centuries take its fourth part and its third part (disregarding fractions in both cases), and increase the result by two. Thus, for the twentieth century we have 20 - 5 - 6 + 2 = 11. Hence, the numerical terms for the next century will be 11 and 26. In old style dates these numerical terms are invariable, and are always two for odd golden numbers and seventeen for even. (See "Proceedings of the Protestant Episcopal Church in the United States" for 1871, Appendix.) The author of this article has also designed an instrumental contrivance for finding Easter by inspection, for any year from the beginning of the Christian era down to the end of hundredth century, in old style or new. This is constructed of card-board, and a facsimile of it, reduced in size, is given below. In the centre is a rotary disk, on the lower border or limb of which are inscribed the numbers below 100 which consist of even twenties, and also the zero. These are called vigesimals. On the upper limb appear all the numbers less than twenty, called residuals, the leap-year numbers being written twice. Around this disk is a fixed ring, bearing the dominical letters above and the centurial numbers below — the new style centurials being on the left, and the old style centurials on the right. The centurial numbers here employed are simply the remainders left in dividing the hundreds by 4 for new style and by 7 for old style. To use this for finding the dominical letter, turn the disk till the proper vigesimal of the given year stands opposite the proper centurial; then opposite the proper residual will be found the dominical letter (or letters) of the year. In case of leap-years there will be found two such letters, of which the lesser or right-hand one is the Easter dominical letter.

Around the fixed ring here described is a rotary ring bearing the numbers from 1 to 19 (the golden numbers), twice repeated, and at the left of these the vigesimals, arranged in regular order. Outside of this rotary ring is a second fixed ring, which bears on the left the numbers 0 to 19, arranged en échelon, so as to allow the natural sequence to be observed. These are called the centurials of the lunar cycle, and are simply what remains after suppressing the nineteens out of the hundreds in the given year of our Lord. Thus, in the year 4173 there are forty-one hundreds, from which, if we suppress 19 × 2 = 38, there will remain 3, which is the centurial for the forty-second century. On the right the same fixed ring bears the residuals, or excesses of twenties in the years of the incomplete century, in which it is not necessary to duplicate the leap-year numbers. When the movable ring is turned so that the proper vigesimal stands opposite the proper centurial, the golden number for the year will be found opposite the proper residual.

On this same fixed ring, outside of the numbers already mentioned, is an annular row of figures distributed without any obvious order, which embraces all the possible golden numbers from 1 to 19, each entered twice. Of these, all up to 11 are printed in full face; all from 12 to 19 inclusive in outline. Their use will presently appear.

Around this second fixed ring is a second rotary ring, on which are inscribed all the days of March and April on which paschal full moon or Easter can fall; together with the calendar letters belonging to them severally. From the 17th to the 25th of April the day numbers and letters are entered twice, the second or inner series being advanced beyond the outer by a single place. This same rotary ring also bears an arrow, which is designed to be used as an index. Finally, surrounding this rotary ring there is another fixed ring, in the several divisions of which are written the centuries from 15 up to 100, none below 15 being necessary, as the new style, or Gregorian reckoning, began in 1582. The use of the last-mentioned rotary ring is to find, first, the date of paschal full moon, and subsequently, by consequence, the date of Easter. In employing it, the ring is turned until the arrow points to the golden number for the year, when the date of paschal full moon will be found opposite the proper centurial number in the outer fixed ring. Then, looking along the series of letters to the right of the date of the paschal moon, Easter will be found immediately over the next succeeding dominical letter for the year. If the time of Easter for years before 1582 is sought, the paschal moon will be found, not opposite the century, but opposite the words "Old Style" written in one of the compartments into which the outer fixed ring is divided, and Easter will be opposite the proper dominical letter next following, as before.

As it is arbitrarily ruled that the paschal full moon shall never fall later than April 18th, and as a consistent method of computation or of instrumental determination would make it sometimes fall on the 19th, the double series of days and letters is introduced at the end of April in the outer revolving ring to meet this case. When, therefore, in the use of the instrument, paschal full moon would seem to fall on the 19th of April by the series of outer, full-faced figures, we must pass to the inner series of figures printed in outline, which will give paschal full moon on the 18th. Also, if the outer series of full-faced figures should at any time directly give paschal full moon on the 18th, we must pass to the inner series again, and make paschal full moon the 17th, provided the arrow stands opposite a golden number printed in outline, but not otherwise. When the light-faced numbers are thus used instead of the full-faced for the paschal moon, the light-faced letters must of course also be used in finding Easter. The table in figure on the preceding page is adjusted for the Easter of 1873. In 73 the vigesimal is 60 and the residual is 13. For 18 (centuries) the centurial is 2, and the Easter sought belongs to new style. It is seen that, 60 being opposite 2, the residual, 13, is opposite E; which is the dominical letter of 1873. In the first rotary ring the same vigesimal, 60, is opposite the golden number centurial, which is 18; and under the residual 13 we have 12, the golden number for 1873. Bringing, finally, the arrow of the outer rotary ring opposite to the golden number, 12, we find under 18 in the outer row of centuries, the 12th of April, which is the date of paschal full moon for 1873; and opposite E, the dominical letter of the year next following the date of the paschal full moon thus found, we have April 13th for the date of Easter.

This little instrument is useful in the solution of many questions connected with chronology and the calendar, besides that for which it was expressly constructed. Any person possessed of a little mechanical skill can construct a working instrument of this kind for himself, by copying this diagram on a scale about one-fourth larger.

The principal festivals and fasts of the Church dependent for the time of their celebration upon Easter are Septuagesima Sunday, nine weeks before Easter; Ash Wednesday, which is the Wednesday of the seventh week before Easter; Good Friday, which is the Friday next before Easter; Ascension Day, which is the Thursday of the sixth week after Easter; Whitsun Day, the seventh Sunday after Easter; and Trinity Sunday, the eighth Sunday after Easter.

F. A. P. Barnard, Columbia College.

Now an attempt at this "easy" (certainly longer than the one-pager he was complaining about) calculation for the year this post was written, 2020:

  1. The article author refers to his year (1873) as being the 18th century. So that would make our year (2020) the 20th century as he calls it. He already calculated the "numerical terms" for the 20th century to be 11 and 26.
  2. The dominical: The hundreds number, 20, is divisible by 4, which gives 0 remainder, so we start with A (1) as our centurial. The years number, 20, is the largest number in itself divisible by 4, and half of that is 10. We increase by seven: 10 + 7 = 17. Adding the centurial and reducing by sevens: 17 + 1 = 18 - 7 = 11 - 7 = 4 (D).
  3. The golden number: Add one to the year: 2020 + 1 = 2021. Taking the last two digits for the years and dividing by four: 21 + 4 = 5 with 1 remainder. Add the remainder to the result: 5 + 1 = 6. Going back to the century and dividing that by four: 20 + 4 = 5. Adding the years and century results together: 5 + 6 = 11.
  4. Our golden number is odd, so we multiply our golden number by four and add the second of the "numerical terms" from step 1: 11 × 4 + 26 = 70. This is greater than 50, so we subtract 30 until we get a number between 20 and 50: 70 - 30 = 40. This is the date of the paschal full moon (March 40 = April 9th).
  5. We then add our dominical number to the constant 18: 18 + 4 = 22. 22 is less than our paschal full moon value of 40, so we keep adding 7 until it exceeds that value: 22 + 7 = 29 + 7 = 36 + 7 = 43. 43 is greater than 40, so Easter is "March 43rd", or April 12th — which is indeed the correct date for 2020.