Popular Science Monthly/Volume 25/August 1884/The Mystic Properties of Numbers
ONE of the earliest French mathematical books is the arithmetic of Étienne de la Roche, in which, the title-page states, are given tables of different accounts, with their canons, calculated by Gilles Huguetan, native of Lyons: "in which may easily be found the accounts all made, as well of purchases as of sales, of all kinds of merchandise. And, principally, of goods which are sold or bought by measure, as by the ell, by the cane, by the toise, by the palm, by the foot, and the like. By weight, as by the pound, by the quintal, by the thousand-weight, by the load, by the half-pound and the ounce, by the piece, by the number, by the dozen, by the gross, by the hundred, and by the thousand. With two tables of use to booksellers, in selling and buying paper, together with a table of expense, showing, at so much a day, how much one spends by the year and the month, and at so much a month how much it comes to by the year and the day, and at so much a year how much one spends in a month and how much it comes to for each day.
"Further, tables of the fineness of gold and silver, showing, according as the coin contains of alloy or fine metal, how much it is worth in the weight of fine gold or of fine silver.
"Sold at Lyons, at the sign of the Sphere, by Gilles and Jaques Huguetan Brothers, 1538."
We give the first chapter of this curious work, which treats of the first twelve numbers, their properties and perfections. Our modern works, while they are less unsophisticated, are certainly far less amusing in expounding the beginnings of arithmetic:
"Number, according to Euclid, at the beginning of the seventh book: Est multitudo ex unitatibus composita (is multitude composed of units). And, again, in the third part of his first book he says: Seriem numero et in infinitum posse procedere ut quocumque numero dato dari potest major unitatem addendo. (A series in number may go on to infinity, so that any given number may be made larger by adding unity.) And in this way number is an aggregation or collection of one or many units. And to proceed in infinitum by the addition of one. From which it appears that unity is not number; but, on the contrary, is the root and foundation of numbers. Even as Boëthius says in his arithmetic. Nevertheless, one is higher and more perfect than all the numbers that are. For in it are united potentially the property and perfection of all numbers. And without it nothing can have being. And Euclid, at the beginning of the seventh book, says: Unitas est qua una quacumque res una dicitur (Unity is that by which any one thing is called one). And the logicians say that one is one of the six transcendent principles. For it comprehends all things that have being. Then, again, it has all the property of number. For it is perfect, like six, it is lineal, square, cube, solid, square root, cube root, root of root. And because it is of so great dignity and excellence, the Creator has chosen it for his essence; for he is one only God, creator of all the world. A good law, to wit: the Christian law, divided into ten commandments. And a good faith: to wit, the Catholic faith, divided into twelve articles. And so many other dignities and perfections.
"Two is a number of so great pre-eminence and utility that God has kept it in mind in many of his works. For first, he created light and darkness. Then he created two great lights, to wit, the sun and the moon. The sun, to light the day; and the moon, to light the night. Then he created all beasts in two sexes, to wit, masculine and feminine; and made for them several double members, to wit, two eyes, two ears, two nostrils, two arms, two hands, two legs, two feet and many others of utility to the human body. And then, as many passions as the human body suffers, such as joy and sadness, hope and fear, hunger and thirst, heat and cold, drinking and eating, sleeping and waking, health and sickness, living and dying, and all relative qualities are also constituted in duplicity, as creator and creature, parent and son, creating and created, producing and produced, abstract and concrete, etc. And also all opposites, as kindness and malice, virtue and vice, knowledge and ignorance, wisdom and foolishness, truth and falsehood, etc. And we think that after unity more things are found constituted by two than by any superior number.
"Three is the most worthy and most perfect, after one, that is among the numbers. Thus, as says almost every one's maxim, Omne trinum perfectum (Every trine is perfect). And the perfection does not proceed by the composition of it, as it does of six. But by the great and high mysteries that are found in this number. And first, it has pleased God, the Creator, to be trine in persons. To wit, Father, Son, and Holy Ghost. It has pleased him to create three hierarchies; and in each hierarchy three orders of angels. There are three things in Jesus Christ, to wit: deity, the soul, and humanity. The priest makes three parts of the precious body of Jesus Christ in the mass. Three holy orders sing the mass, to wit: the priest, the deacon, and the sub-deacon. Three times are sung the Sanctus, Sanctus Sanctus; and the Agnus Dei qui tollis peccata in the mass. By three nails was fastened the Blessed Redeemer Jesus Christ on the cross. There are three degrees of penitence, to wit: contrition, confession, and satisfaction. There are three parts of satisfaction, to wit: fasting, alms, and prayer. There are three divine virtues, to wit: faith, hope, and charity. There are three enemies of the soul, to wit: the world, the flesh, and the devil. Man sins in three ways, to wit: with heart, with word, and with deed. Man may offend three things, to wit: God, himself, and his neighbor. God has disposed all things by number, by weight, and by measure. Tria erant in archa; s. c. g. Virga; manna et lex Mosayca (there were three things in the ark: the rod, the manna, and the Mosaic law). Three places are deputed for man after his death, to wit: paradise, purgatory, and hell. Three vows do the minor friars vow when they make profession, to wit: poverty, obedience, and chastity. There are three natural principles, to wit: form, matter, and privation; or, potentia (power); objectum (object); et actus (and act). There are three souls, to wit: vegetative, sensitive, and rational. There are three powers in the rational soul, to wit: will, memory, and understanding. Bodies have three dimensions, to wit: length, breadth, and thickness. The world is divided into three parts, to wit: into Asia, Europe, and Africa. And thus appears the excellence and magnificence of this worthy number three.
"Four is the first square number, and is of great esteem and necessity. And first, God, the Creator, has created four elements, to wit: fire, air, water, and earth. Whence proceed four qualities, to wit: warmth, frigidity, dryness, and moisture. From which arise four humors, to wit: blood, bile, phlegm, and melancholy; by which are caused four complexions, to wit: sanguine, choleric, phlegmatic, and melancholic. There are four seasons in the year, to wit: spring, summer, autumn, and winter. And four quarters in the sky and the world, to wit: eastern, western, northern, and southern. And to each quarter one principal wind, to wit: the morning or east-wind, the traverse or west-wind, the north-wind or transmontane, and the sea or south-wind. And, according to the philosophers, there are four causes in all things, to wit: the efficient, formal, material, and final causes. There are four cardinal virtues, to wit: prudence, temperance, strength, and justice. The glorified body in paradise has four endowments, to wit: brightness, subtilty, agility, and impassibility. There are four evangelists to certify the faith of Jesus Christ. And four principal doctors of the Church to corroborate the faith, to wit: St. Augustine, Gregory, Hieronymus, and St. Ambrose.
"Five is a number of great convenience and utility; for, first, the Creator created five simple bodies, to wit: the sky, fire, air, water, and earth. And never have more of regular bodies that have equal bases been found. Then, for our use, the Creator has given us five natural senses, to wit: sight, hearing, taste, touch, and smell. And five fingers on the hand, and five toes on the foot. And to redeem us has suffered five wounds on the cross, and in all the surface of the earth there are five zones, according to Sacrobosco in his sphere.
"Six is the first and most worthy of the perfect numbers. Because in its composition three aliquot parts put together make their whole: as 3, 2, and 1, which are its 1⁄2, its 1⁄3, and its 1⁄6. Which put together amount to 6, which is their whole. There is another perfection, because it is a circular number. For, in making a circle, with a compass, the circumference of the circle contains just six times the span of the compass; as when one should put one of the feet of the said compass on the circumference of the said circle, and should turn the said compass to six times on the said circumference. At the sixth time the said foot of the compass would return to its first point. And because it returns always in itself. Et semperidem ipse est (and it is always the same). There is also another perfection, because there are six transcendent principles, to wit: one, good, true, thing, something, and being. And for these great perfections and dignities, the Creator regards it in his works, for he created everything in six days. Therefore ought it to be named the very perfect among the perfect numbers. So has St. Augustine said in the thirtieth chapter of the second book 'De Civitate Dei.'
"Seven is a number of great prerogative and singularity, as St. Augustine says in the thirty-first chapter of the aforesaid book. Because of its composition which is triple, for first it is composed of 1 and of 6, which are of so great perfection as is said above. Or of 3 and 4, which are of so great dignity and estimation. Or of 2 and 5, which are of so great utility and commodity. And because in its composition it contains so many numbers worthy, perfect, and of great excellence. God, the Creator, regards it in his most admirable works. For he has created seven planets, seven metals, seven colors, and seven tastes. And when he had created everything in six days, he rested on the seventh, which is a thing of great mystery. There are, therefore, seven days in the week. There are seven principal virtues, to wit: three divine, and four cardinal. There are seven other virtues against the seven mortal sins. There are seven works of bodily mercy and seven works of spiritual mercy. There are seven sacraments. There are seven orders in the holy church. There are seven ages of man. There are seven windows through which the ordinary senses are exercised: the two eyes, the two ears, the two nostrils, and the mouth. There are seven days between the setting in of a disease and the critical day. There are seven climates in the habitable earth.
"Eight is the first cube number, and there are also eight beatitudes.
"Nine is the second square number, and there are also nine orders of angels and the kirie eleison is sung nine times in the mass.
"Ten, according to some, is a perfect number, not in its composition like six, but because it contains inclosed in itself all the simple numbers and all the properties, as of even and odd, perfect and imperfect, and it is the beginning of all numbers composed of tens, and also for the foundation of our law. God gave to Moses the ten commandments of the law, and ordered men to give to God the tithe, which is one tenth of his gain or his labor.
"Eleven is the first compound odd number.
"Twelve is a number of great pre-eminence and utility. And, although it is an excessively imperfect number, it is nevertheless of great utility. For it can, first, be divided into more parts than any number below it. For it can be evenly divided by 6, which is its 1⁄2; by 4, which is its 1⁄3; by 3, which is its 1⁄4; and by 2, which is its 1⁄6. And because the blessed Redeemer Jesus Christ wished to observe the said number in choosing twelve apostles to found and form the Holy Catholic Faith. Who for the foundation of the same composed the twelve articles of the faith. And in imitation of them the lords of the cathedral churches constitute twelve perpetuals to listen continually to the service of God the Creator; twelve choralists to sing the hymns of God and the saints. Likewise, on account of the convenience of this number, the good governors of cities commonly choose twelve counselors to attend to the regulation of the public good. So the astrological philosophers of the ancient times, experimenting and considering the celestial natures and influences, divided the whole sky into twelve equal parts which were called the twelve signs. And they attributed to each a peculiar influence by subtile commixtion, and established twelve months in the year for greater convenience. And this is enough of the property of the numbers in particular.—Translated for the Popular Science Monthly from the Revue Scientifique.