Tuesday, 28 March 2017

Marcus Vitruvius Pollio - Book IX of De Architecture

"Marcus Vitruvius Pollio (born c. 80–70 BC, died after c. 15 BC), commonly known as Vitruvius or Vitruvi or Vitruvio, was a Roman author, architect, civil engineer and military engineer during the 1st century BC, known for his multi-volume work entitled De architectura. His discussion of perfect proportion in architecture and the human body, led to the famous Renaissance drawing by Da Vinci of Vitruvian Man." 


English translation by Morris Morgan (1914).  Headings are changed to reflect the chapters in the Latin versions. Images are from several 16th century Latin publications (1511, 1523, 1567).






1. The ancestors of the Greeks have appointed such great honours for the famous athletes who are victorious at the Olympian, Pythian, Isthmian, and Nemean games, that they are not only greeted with applause as they stand with palm and crown at the meeting itself, but even on returning to their several states in the triumph of victory, they ride into their cities and to their fathers' houses in four-horse chariots, and enjoy fixed revenues for life at the public expense. When I think of this, I am amazed that the same honours and even greater are not bestowed upon those authors whose boundless services are performed for all time and for all nations. This would have been a practice all the more worth establishing, because in the case of athletes it is merely their own bodily frame that is strengthened by their training, whereas in the case of authors it is the mind, and not only their own but also man's in general, by the doctrines laid down in their books for the acquiring of knowledge and the sharpening of the intellect.

2. What does it signify to mankind that Milo of Croton and other victors of his class were invincible? Nothing, save that in their lifetime they were famous among their countrymen. But the doctrines of Pythagoras, Democritus, Plato, and Aristotle, and the daily life of other learned men, spent in constant industry, yield fresh and rich fruit, not only to their own countrymen, but also to all nations. And they who from their tender years are filled with the plenteous learning which this fruit affords, attain to the highest capacity of knowledge, and can introduce into their states civilized ways, impartial justice, and laws, things without which no state can be sound. 

3. Since, therefore, these great benefits to individuals and to communities are due to the wisdom of authors, I think that not only should palms and crowns be bestowed upon them, but that they should even be granted triumphs, and judged worthy of being consecrated in the dwellings of the gods

Of their many discoveries which have been useful for the development of human life, I will cite a few examples. On reviewing these, people will admit that honours ought of necessity to be bestowed upon them.


1. First of all, among the many very useful theorems of Plato, I will cite one as demonstrated by him. Suppose there is a place or a field in the form of a square and we are required to double it. This has to be effected by means of lines correctly drawn, for it will take a kind of calculation not to be made by means of mere multiplication. The following is the demonstration. A square place ten feet long and ten feet wide gives an area of one hundred feet. Now it is required to double the square, and to make one of two hundred feet, we must ask how long will be the side of that square so as to get from this the two hundred feet corresponding to the doubling of the area. Nobody can find this by means of arithmetic. For if we take fourteen, multiplication will give one hundred and ninety-six feet; if fifteen, two hundred and twenty-five feet.

2. Therefore, since this is inexplicable by arithmetic, let a diagonal line be drawn from angle to angle of that square of ten feet in length and width, dividing it into two triangles of equal size, each fifty feet in area. Taking this diagonal line as the length, describe another square. Thus we shall have in the larger square four triangles of the same size and the same number of feet as the two of fifty feet each which were formed by the diagonal line in the smaller square. In this way Plato demonstrated the doubling by means of lines, as the figure appended below will show.


1. Then again, Pythagoras showed that a right angle can be formed without the contrivances of the artisan. Thus, the result which carpenters reach very laboriously, but scarcely to exactness, with their squares, can be demonstrated to perfection from the reasoning and methods of his teaching. If we take three rules, one three feet, the second four feet, and the third five feet in length, and join these rules together with their tips touching each other so as to make a triangular figure, they will form a right angle. Now if a square be described on the length of each one of these rules, the square on the side of three feet in length will have an area of nine feet; of four feet, sixteen; of five, twenty-five.

2. Thus the area in number of feet made up of the two squares on the sides three and four feet in length is equalled by that of the one square described on the side of five. When Pythagoras discovered this fact, he had no doubt that the Muses had guided him in the discovery, and it is said that he very gratefully offered sacrifice to them.  This theorem affords a useful means of measuring many things, and it is particularly serviceable in the building of staircases in buildings, so that the steps may be at the proper levels.

3. Suppose the height of the story, from the flooring above to the ground below, to be divided into three parts. Five of these will give the right length for the stringers of the stairway. Let four parts, each equal to one of the three composing the height between the upper story and the ground, be set off from the perpendicular, and there fix the lower ends of the stringers. In this manner the steps and the stairway itself will be properly placed. A figure of this also will be found appended below.


1. In the case of Archimedes, although he made many wonderful discoveries of diverse kinds, yet of them all, the following, which I shall relate, seems to have been the result of a boundless ingenuity. Hiero, after gaining the royal power in Syracuse, resolved, as a consequence of his successful exploits, to place in a certain temple a golden crown which he had vowed to the immortal gods. He contracted for its making at a fixed price, and weighed out a precise amount of gold to the contractor. At the appointed time the latter delivered to the king's satisfaction an exquisitely finished piece of handiwork, and it appeared that in weight the crown corresponded precisely to what the gold had weighed.

2. But afterwards a charge was made that gold had been abstracted and an equivalent weight of silver had been added in the manufacture of the crown. Hiero, thinking it an outrage that he had been tricked, and yet not knowing how to detect the theft, requested Archimedes to consider the matter. The latter, while the case was still on his mind, happened to go to the bath, and on getting into a tub observed that the more his body sank into it the more water ran out over the tub. As this pointed out the way to explain the case in question, without a moment's delay, and transported with joy, he jumped out of the tub and rushed home naked, crying with a loud voice that he had found what he was seeking; for as he ran he shouted repeatedly in Greek, "Ευρηκα, ευρηκα."

3. Taking this as the beginning of his discovery, it is said that he made two masses of the same weight as the crown, one of gold and the other of silver. After making them, he filled a large vessel with water to the very brim, and dropped the mass of silver into it. As much water ran out as was equal in bulk to that of the silver sunk in the vessel. Then, taking out the mass, he poured back the lost quantity of water, using a pint measure, until it was level with the brim as it had been before. Thus he found the weight of silver corresponding to a definite quantity of water.

4. After this experiment, he likewise dropped the mass of gold into the full vessel and, on taking it out and measuring as before, found that not so much water was lost, but a smaller quantity: namely, as much less as a mass of gold lacks in bulk compared to a mass of silver of the same weight. Finally, filling the vessel again and dropping the crown itself into the same quantity of water, he found that more water ran over for the crown than for the mass of gold of the same weight. Hence, reasoning from the fact that more water was lost in the case of the crown than in that of the mass, he detected the mixing of silver with the gold, and made the theft of the contractor perfectly clear.

5. Now let us turn our thoughts to the researches of Archytas of Tarentum and Eratosthenes of Cyrene. They made many discoveries from mathematics which are welcome to men, and so, though they deserve our thanks for other discoveries, they are particularly worthy of admiration for their ideas in that field. For example, each in a different way solved the problem enjoined upon Delos by Apollo in an oracle, the doubling of the number of cubic feet in his altars; this done, he said, the inhabitants of the island would be delivered from an offence against religion.

6. Archytas solved it by his figure of the semicylinders; Eratosthenes, by means of the instrument called the mesolabe. 

Noting all these things with the great delight which learning gives, we cannot but be stirred by these discoveries when we reflect upon the influence of them one by one. I find also much for admiration in the books of Democritus on nature, and in his commentary entitled Χειρὁκμητα, in which he made use of his ring to seal with soft wax the principles which he had himself put to the test.

7. These, then, were men whose researches are an everlasting possession, not only for the improvement of character but also for general utility. The fame of athletes, however, soon declines with their bodily powers. Neither when they are in the flower of their strength, nor afterwards with posterity, can they do for human life what is done by the researches of the learned. 

8. But although honours are not bestowed upon authors for excellence of character and teaching, yet as their minds, naturally looking up to the higher regions of the air, are raised to the sky on the steps of history, it must needs be, that not merely their doctrines, but even their appearance, should be known to posterity through time eternal. Hence, men whose souls are aroused by the delights of literature cannot but carry enshrined in their hearts the likeness of the poet Ennius, as they do those of the gods. Those who are devotedly attached to the poems of Accius seem to have before them not merely his vigorous language but even his very figure. 

9. So, too, numbers born after our time will feel as if they were discussing nature face to face with Lucretius, or the art of rhetoric with Cicero; many of our posterity will confer with Varro on the Latin language; likewise, there will be numerous scholars who, as they weigh many points with the wise among the Greeks, will feel as if they were carrying on private conversations with them. Li a word, the opinions of learned authors, though their bodily forms are absent, gain strength as time goes on, and, when taking part in councils and discussions, have greater weight than those of any living men.
10. Such, Caesar, are the authorities on whom I have depended, and applying their views and opinions I have written the present books, in the first seven treating of buildings and in the eighth of water. In this I shall set forth the rules for dialling, showing how they are found through the shadows cast by the gnomon from the sun's rays in the firmament, and on what principles these shadows lengthen and shorten. 


1. It is due to the divine intelligence and is a very great wonder to all who reflect upon it, that the shadow of a gnomon at the equinox is of one length in Athens, of another in Alexandria, of another in Rome, and not the same at Piacenza, or at other places in the world. Hence drawings for dials are very different from one another, corresponding to differences of situation. This is because the length of the shadow at the equinox is used in constructing the figure of the analemma, in accordance with which the hours are marked to conform to the situation and the shadow of the gnomon. The analemma is a basis for calculation deduced from the course of the sun, and found by observation of the shadow as it increases until the winter solstice. By means of this, through architectural principles and the employment of the compasses, we find out the operation of the sun in the universe. 

2. The word "universe" means the general assemblage of all nature, and it also means the heaven that is made up of the constellations and the courses of the stars. The heaven revolves steadily round earth and sea on the pivots at the ends of its axis. The architect at these points was the power of Nature, and she put the pivots there, to be, as it were, centres, one of them above the earth and sea at the very top of the firmament and even beyond the stars composing the Great Bear, the other on the opposite side under the earth in the regions of the south. Round these pivots (termed in Greek πόλοι) as centres, like those of a turning lathe, she formed the circles in which the heaven passes on its everlasting way. In the midst thereof, the earth and sea naturally occupy the central point.

3. It follows from this natural arrangement that the central point in the north is high above the earth, while on the south, the region below, it is beneath the earth and consequently hidden by it. Furthermore, across the middle, and obliquely inclined to the south, there is a broad circular belt composed of the twelve signs, whose stars, arranged in twelve equivalent divisions, represent each a shape which nature has depicted. And so with the firmament and the other constellations, they move round the earth and sea in glittering array, completing their orbits according to the spherical shape of the heaven.

4. They are all visible or invisible according to fixed times. While six of the signs are passing along with the heaven above the earth, the other six are moving under the earth and hidden by its shadow. But there are always six of them making their way above the earth; for, corresponding to that part of the last sign which in the course of its revolution has to sink, pass under the earth, and become concealed, an equivalent part of the sign opposite to it is obliged by the law of their common revolution to pass up and, having completed its circuit, to emerge out of the darkness into the light of the open space on the other side. This is because the rising and setting of both are subject to one and the same power and law.

5. While these signs, twelve in number and occupying each one twelfth part of the firmament, steadily revolve from east to west, the moon. Mercury, Venus, the sun, as well as Mars, Jupiter, and Saturn, differing from one another in the magnitude of their orbits as though their courses were at different points in a flight of steps, pass through those signs in just the opposite direction, from west to east in the firmament. The moon makes her circuit of the heaven in twenty-eight days plus about an hour, and with her return to the sign from which she set forth, completes a lunar month.

6. The sun takes a full month to move across the space of one sign, that is, one twelfth of the firmament. Consequently, in twelve months he traverses the spaces of the twelve signs, and, on returning to the sign from which he began, completes the period of a full year. Hence, the circuit made by the moon thirteen times in twelve months, is measured by the sun only once in the same number of months. But Mercury and Venus, their paths wreathing around the sun's rays as their centre, retrograde and delay their movements, and so, from the nature of that circuit, sometimes wait at stopping-places within the spaces of the signs.

7. This fact may best be recognized from Venus. When she is following the sun, she makes her appearance in the sky after his setting, and is then called the Evening Star, shining most brilliantly. At other times she precedes him, rising before daybreak, and is named the Morning Star. Thus Mercury and Venus sometimes delay in one sign for a good many days, and at others advance pretty rapidly into another sign. They do not spend the same number of days in every sign, but the longer they have previously delayed, the more rapidly they accomplish their journeys after passing into the next sign, and thus they complete their appointed course. Consequently, in spite of their delay in some of the signs, they nevertheless soon reach the proper place in their orbits after freeing themselves from their enforced delay.

8. Mercury, on his journey through the heavens, passes through the spaces of the signs in three hundred and sixty days, and so arrives at the sign from which he set out on his course at the beginning of his revolution. His average rate of movement is such that he has about thirty days in each sign.

9. Venus, on becoming free from the hindrance of the sun's rays, crosses the space of a sign in thirty days. Though she thus stays less than forty days in particular signs, she makes good the required amount by delaying in one sign when she comes to a pause. Therefore she completes her total revolution in heaven in four hundred and eighty-five days, and once more enters the sign from which she previously began to move.

10. Mars, after traversing the spaces of the constellations for about six hundred and eighty-three days, arrives at the point from which he had before set out at the beginning of his course, and while he passes through some of the signs more rapidly than others, he makes up the required number of days whenever he comes to a pause. Jupiter, climbing with gentler pace against the revolution of the firmament, travels through each sign in about three hundred and sixty days, and finishes in eleven years and three hundred and thirteen days, returning to the sign in which he had been twelve years before. Saturn, traversing the space of one sign in twenty-nine months plus a few days, is restored after twenty-nine years and about one hundred and sixty days to that in which he had been thirty years before. He is, as it appears, slower, because the nearer he is to the outermost part of the firmament, the greater is the orbit through which he has to pass.

11. The three that complete their circuits above the sun's course do not make progress while they are in the triangle which he has entered, but retrograde and pause until the sun has crossed from that triangle into another sign. Some hold that this takes place because, as they say, when the sun is a great distance off, the paths on which these stars wander are without light on account of that distance, and so the darkness retards and hinders them. But I do not think that this is so. The splendour of the sun is clearly to be seen, and manifest without any kind of obscurity, throughout the whole firmament, so that those very retrograde movements and pauses of the stars are visible even to us.

12. If then, at this great distance, our human vision can discern that sight, why, pray, are we to think that the divine splendour of the stars can be cast into darkness? Rather will the following way of accounting for it prove to be correct. Heat summons and attracts everything towards itself; for instance, we see the fruits of the earth growing up high under the influence of heat, and that spring water is vapourised and drawn up to the clouds at sunrise. On the same principle, the mighty influence of the sun, with his rays diverging in the form of a triangle, attracts the stars which follow him, and, as it were, curbs and restrains those that precede, not allowing them to make progress, but obliging them to retrograde towards himself until he passes out into the sign that belongs to a different triangle.

13. Perhaps the question will be raised, why the sun by his great heat causes these detentions in the fifth sign from himself rather than in the second or third, which are nearer. I will therefore set forth what seems to be the reason. His rays diverge through the firmament in straight lines as though forming an equilateral triangle, that is, to the fifth sign from the sun, no more, no less. If his rays were diffused in circuits spreading all over the firmament, instead of in straight lines diverging so as to form a triangle, they would burn up all the nearer objects. This is a fact which the Greek poet Euripides seems to have remarked; for he says that places at a greater distance from the sun are in a violent heat, and that those which are nearer he keeps temperate. Thus in the play of Phaethon, the poet writes: καἱει τἁ πὁρρω, τἁγγυθεν δ εὑκρατ ἑχει.

14. If then, fact and reason and the evidence of an ancient poet point to this explanation, I do not see why we should decide otherwise than as I have written above on this subject. Jupiter, whose orbit is between those of Mars and Saturn, traverses a longer course than Mars, and a shorter than Saturn. Likewise with the rest of these stars : the farther they are from the outermost limits of the heaven, and the nearer their orbits to the earth, the sooner they are seen to finish their courses; for those of them that have a smaller orbit often pass those that are higher, going under them.

15. For example, place seven ants on a wheel such as potters use, having made seven channels on the wheel about the centre, increasing successively in circumference; and suppose those ants obliged to make a circuit in these channels while the wheel is turned in the opposite direction. In spite of having to move in a direction contrary to that of the wheel, the ants must necessarily complete their journeys in the opposite direction, and that ant which is nearest the centre must finish its circuit sooner, while the ant that is going round at the outer edge of the disc of the wheel must, on account of the size of its circuit, be much slower in completing its course, even though it is moving just as quickly as the other. In the same way, these stars, which struggle on against the course of the firmament, are accomplishing an orbit on paths of their own; but, owing to the revolution of the heaven, they are swept back as it goes round every day.

16. The reason why some of these stars are temperate, others hot, and others cold, appears to be this: that the flame of every kind of fire rises to higher places. Consequently, the burning rays of the sun make the ether above him white hot, in the regions of the course of Mars, and so the heat of the sun makes him hot. Saturn, on the contrary, being nearest to the outermost limit of the firmament and bordering on the quarters of the heaven which are frozen, is excessively cold. Hence Jupiter, whose course is between the orbits of these two, appears to have a moderate and very temperate influence, intermediate between their cold and heat.

I have now described, as I have received them from my teacher, the belt of the twelve signs and the seven stars that work and move in the opposite direction, with the laws and numerical relations under which they pass from sign to sign, and how they complete their orbits. I shall next speak of the waxing and waning of the moon, according to the accounts of my predecessors.


1. According to the teaching of Berosus, who came from the state, or rather nation, of the Chaldees, and was the pioneer of Chaldean learning in Asia, the moon is a ball, one half luminous and the rest of a blue colour. When, in the course of her orbit, she has passed below the disc of the sun, she is attracted by his rays and great heat, and turns thither her luminous side, on account of the sympathy between light and light. Being thus summoned by the sun's disc and facing upward, her lower half, as it is not  luminous, is invisible on account of its likeness to the air. When she is perpendicular to the sun's rays, all her light is confined to her upper surface, and she is then called the new moon.

2. As she moves on, passing by to the east, the effect of the sun upon her relaxes, and the outer edge of the luminous side sheds its light upon the earth in an exceedingly thin line. This is called the second day of the moon. Day by day she is further relieved and turns, and thus are numbered the third, fourth, and following days. On the seventh day, the sun being in the west and the moon in the middle of the firmament between the east and west, she is half the extent of the firmament distant from the sun, and therefore half of the luminous side is turned toward the earth. But when the sun and moon are separated by the entire extent of the firmament, and the moon is in the east with the sun over against her in the west, she is completely relieved by her still greater distance from his rays, and so, on the fourteenth day, she is at the full, and her entire disc emits its light. On the succeeding days, up to the end of the month, she wanes daily as she turns in her course, being recalled by the sun until she comes under his disc and rays, thus completing the count of the days of the month.

3. But Aristarchus of Samos, a mathematician of great powers, has left a different explanation in his teaching on this subject, as I shall now set forth. It is no secret that the moon has no light of her own, but is, as it were, a mirror, receiving brightness from the influence of the sun. Of all the seven stars, the moon traverses the shortest orbit, and her course is nearest to the earth. Hence in every month, on the day before she gets past the sun, she is under his disc and rays, and is consequently hidden and invisible. When she is thus in conjunction with the sun, she is called the new moon. On the next day, reckoned as her second, she gets past the sun and shows the thin edge of her sphere. Three days away from the sun, she waxes and grows brighter. Removing further every day till she reaches the seventh, when her distance from the sun at his setting is about one half the extent of the firmament, one half of her is luminous; that is, the half which faces toward the sun is lighted up by him.

4. On the fourteenth day, being diametrically across the whole extent of the firmament from the sun, she is at her full and rises when the sun is setting. For, as she takes her place over against him and distant the whole extent of the firmament, she thus receives the light from the sun throughout her entire orb. On the seventeenth day, at sunrise, she is inclining to the west. On the twenty-second day, after sunrise, the moon is about mid-heaven; hence, the side exposed to the sun is bright and the rest dark. Continuing thus her daily course, she passes under the rays of the sun on about the twenty-eighth day, and so completes the account of the month.

I will next explain how the sun, passing through a different sign each month, causes the days and hours to increase and diminish in length.


1. The sun, after entering the sign Aries and passing through one eighth of it, determines the vernal equinox. On reaching the tail of Taurus and the constellation of the Pleiades, from which the front half of Taurus projects, he advances into a space greater than half the firmament, moving toward the north. From Taurus he enters Gemini at the time of the rising of the Pleiades, and, getting higher above the earth, he increases the length of the days. Next, coming from Gemini into Cancer, which occupies the shortest space in heaven, and after traversing one eighth of it, he determines the summer solstice. Continuing on, he reaches the head and breast of Leo, portions which are reckoned as belonging to Cancer.

2. After leaving the breast of Leo and the boundaries of Cancer, the sun, traversing the rest of Leo, makes the days shorter, diminishing the size of his circuit, and returning to the same course that he had in Gemini. Next, crossing from Leo into Virgo, and advancing as far as the bosom of her garment, he still further shortens his circuit, making his course equal to what it was in Taurus. Advancing from Virgo by way of the bosom of her garment, which forms the first part of Libra, he determines the autumn equinox at the end of one eighth of Libra. Here his course is equal to what his circuit was in the sign Aries.

3. When the sun has entered Scorpio, at the time of the setting of the Pleiades, he begins to make the days shorter as he advances toward the south. From Scorpio he enters Sagittarius and, on reaching the thighs, his daily course is still further diminished. From the thighs of Sagittarius, which are reckoned as part of Capricornus, he reaches the end of the first eighth of the latter, where his course in heaven is shortest. Consequently, this season, from the shortness of the day, is called bruma or dies brumales. Crossing from Capricornus into Aquarius, he causes the days to increase to the length which they had when he was in Sagittarius. From Aquarius he enters Pisces at the time when Favonius begins to blow, and here his course is the same as in Scorpio. Li this way the sun passes round through the signs, lengthening or shortening the days and hours at definite seasons.

I shall next speak of the other constellations formed by arrangements of stars, and lying to the right and left of the belt of the signs, in the southern and northern portions of the firmament.


1. The Great Bear, called in Greek ἁρκτος or ἑλἱκη, has her Warden behind her. Near him is the Virgin, on whose right shoulder rests a very bright star which we call Harbinger of the Vintage, and the Greeks,  προτρυγητἡς. But Spica in that constellation is brighter. Opposite there is another star, coloured, between the knees of the Bear Warden, dedicated there under the name of Arcturus.

2. Opposite the head of the Bear, at an angle with the feet of the Twins, is the Charioteer, standing on the tip of the horn of the Bull; hence, one and the same star is found in the tip of the left horn of the Bull and in the right foot of the Charioteer. Supported on the hand of the Charioteer are the Kids, with the She-Goat at his left shoulder. Above the Bull and the Ram is Perseus, having at his right .with the Pleiades moving beneath, and at his left the head of the Ram. His right hand rests on the likeness of Cassiopea, and with his left he holds the Gorgon's head by its top over the Ram, laying it at the feet of Andromeda.

3. Above Andromeda are the Fishes, one above her belly and the other above the backbone of the Horse. A very bright star terminates both the belly of the Horse and the head of Andromeda. Andromeda's right hand rests above the likeness of Cassiopea, and her left above the Northern Fish. The Waterman's head is above that of the Horse. The Horse's hoofs lie close to the Waterman's knees. Cassiopea is set apart in the midst. High above the He-Goat are the Eagle and the Dolphin, and near them is the Arrow. Farther on is the Bird, whose right wing grazes the head and sceptre of Cepheus, with its left resting over Cassiopea. Under the tail of the Bird lie the feet of the Horse.

4. Above the Archer, Scorpion, and Balance, is the Serpent, reaching to the Crown with the end of its snout. Next, the Serpent-holder grasps the Serpent about the middle in his hands, and with his left foot treads squarely on the foreparts of the Scorpion. A little way from the head of the Serpent-holder is the head of the so-called Kneeler. Their heads are the more readily to be distinguished as the stars which compose them are by no means dim.

5. The foot of the Kneeler rests on the temple of that Serpent which is entwined between the She-Bears (called Septentriones). The little Dolphin moves in front of the Horse. Opposite the bill of the Bird is the Lyre. The Crown is arranged between the shoulders of the Warden and the Kneeler. In the northern circle are the two She-Bears with their shoulder-blades confronting and their breasts turned away from one another. The Greeks call the Lesser Bear κυνὁσουρα, and the Greater ἑλικη. Their heads face different ways, and their tails are shaped so that each is in front of the head of the other Bear; for the tails of both stick up over them.

6. The Serpent is said to lie stretched out between their tails, and in it there is a star, called Polus, shining near the head of the Greater Bear. At the nearest point, the Serpent winds its head round, but is also flung in a fold round the head of the Lesser Bear, and stretches out close to her feet. Here it twists back, making another fold, and, lifting itself up, bends its snout and right temple from the head of the Lesser Bear round towards the Greater. Above the tail of the Lesser Bear are the feet of Cepheus, and at this point, at the very top, are stars forming an equilateral triangle. There are a good many stars common to the Lesser Bear and to Cepheus.

I have now mentioned the constellations which are arranged in the heaven to the right of the east, between the belt of the signs and the north. I shall next describe those that Nature has distributed to the left of the east and in the southern regions. 


1. First, under the He-Goat lies the Southern Fish, facing towards the tail of the Whale. The Altar is under the Scorpion's sting. The fore parts of the Centaur are next to the Balance and the Scorpion, and he holds in his hands the figure which astronomers call the Beast. Beneath the Virgin, Lion, and Crab is the twisted girdle formed by the Snake, extending over a whole line of stars, his snout raised near the Crab, supporting the Bowl with the middle of his body near the Lion, and bringing his tail, on which is the Raven, under and near the hand of the Virgin. The region above his shoulders is equally bright.

2. Beneath the Snake's belly, at the tail, lies the Centaur. Near the Bowl and the Lion is the ship named Argo. Her bow is invisible, but her mast and the parts about the helm are in plain sight, the stern of the vessel joining the Dog at the tip of his tail. The Little Dog follows the Twins, and is opposite the Snake's head. The Greater Dog follows the Lesser. Orion lies aslant, under the Bull's hoof; in his left hand grasping his club, and raising the other toward the Twins.

3. At his feet is the Dog, following a little behind the Hare. The Whale lies under the Ram and the Fishes, and from his mane there is a slight sprinkling of stars, called in Greek ἁρπεδὁναι, regularly disposed towards each of the Fishes. This ligature by which they hang is carried a great way inwards, but reaches out to the top of the mane of the Whale. The River, formed of stars, flows from a source at the left foot of Orion. But the Water, said to pour from the Waterman, flows between the head of the Southern Fish and the tail of the Whale.

4. These constellations, whose outlines and shapes in the heavens were designed by Nature and the divine intelligence, I have described according to the view of the natural philosopher Democritus, but only those whose risings and settings we can observe and see with our own eyes. Just as the Bears turn round the pivot of the axis without ever setting or sinking under the earth, there are likewise stars that keep turning round the southern pivot, which on account of the inclination of the firmament lies always under the earth, and, being hidden there, they never rise and emerge above the earth. Consequently, the figures which they form are unknown to us on account of the interposition of the earth. The star Canopus proves this. It is unknown to our vicinity; but we have reports of it from merchants who have been to the most distant part of Egypt, and to regions bordering on the uttermost boundaries of the earth.


1. I HAVE shown how the firmament, and the twelve signs with the constellations arranged to the north and south of them, fly round the earth, so that the matter may be clearly understood. For it is from this revolution of the firmament, from the course of the sun through the signs in the opposite direction, and from the shadows cast by equinoctial gnomons, that we find the figure of the analemma.

2. As for the branch of astronomy which concerns the infiuences of the twelve signs, the five stars, the sun, and the moon upon human life, we must leave all this to the calculations of the Chaldeans, to whom belongs the art of casting nativities, which enables them to declare the past and the future by means of calculations based on the stars. These discoveries have been transmitted by the men of genius and great acuteness who sprang directly from the nation of the Chaldeans; first of all, by Berosus, who settled in the island state of Cos, and there opened a school. Afterwards Antipater pursued the subject; then there was Archinapolus, who also left rules for casting nativities, based not on the moment of birth but on that of conception.

3. When we come to natural philosophy, however, Thales of Miletus, Anaxagoras of Clazomenae, Pythagoras of Samos,Xenophanes of Colophon, and Democritus of Abdera have in various ways investigated and left us the laws and the working of the laws by which nature governs it. In the track of their discoveries, Eudoxus, Euctemon, Callippus, Meto, Philippus, Hipparchus, Aratus, and others discovered the risings and settings of the constellations, as well as weather prognostications from astronomy through the study of the calendars, and this study they set forth and left to posterity. Their learning deserves the admiration of mankind; for they were so solicitous as even to be able to predict, long beforehand, with divining mind, the signs of the weather which was to follow in the future. On this subject, therefore, reference must be made to their labours and investigations. 


1. In distinction from the subjects first mentioned, we must ourselves explain the principles which govern the shortening and lengthening of the day. When the sun is at the equinoxes, that is, passing through Aries or Libra, he makes the gnomon cast a shadow equal to eight ninths of its own length, in the latitude of Rome. In Athens, the shadow is equal to three fourths of the length of the gnomon; at Rhodes to five sevenths; at Tarentum, to nine elevenths; at Alexandria, to three fifths; and so at other places it is found that the shadows of equinoctial gnomons are naturally different from one another.

2. Hence, wherever a sundial is to be constructed, we must take the equinoctial shadow of the place. If it is found to be, as in Rome, equal to eight ninths of the gnomon, let a line be drawn on a plane surface, and in the middle thereof erect a perpendicular, plumb to the line, which perpendicular is called the gnomon. Then, from the line in the plane, let the line of the gnomon be divided off by the compasses into nine parts, and take the point designating the ninth part as a centre, to be marked by the letter A. Then, opening the compasses from that centre to the line in the plane at the point B, describe a circle. This circle is called the meridian. 

3. Then, of the nine parts between the plane and the centre on the gnomon, take eight, and mark them off on the line in the plane to the point C. This will be the equinoctial shadow of the gnomon. From that point, marked by C, let a line be drawn through the centre at the point A, and this will represent a ray of the sun at the equinox. Then, extending the compasses from the centre to the line in the plane, mark off the equidistant points E on the left and I on the right, on the two sides of the circumference, and let a line be drawn through the centre, dividing the circle into two equal semicircles. This line is called by mathematicians the horizon.

4. Then, take a fifteenth part of the entire circumference, and, placing the centre of the compasses on the circumference at the point where the equinoctial ray cuts it at the letter F, mark off the points G and H on the right and left. Then lines must be drawn from these (and the centre) to the line of the plane at the points T and R, and thus, one will represent the ray of the sun in winter, and the other the ray in summer. Opposite E will be the point I, where the line drawn through the centre at the point A cuts the circumference; opposite G and H will be the points L and K; and opposite C, F, and A will be the point N.

5. Then, diameters are to be drawn from G to L and from H to K. The upper will denote the summer and the lower the winter portion. These diameters are to be divided equally in the middle at the points M and O, and those centres marked; then, through these marks and the centre A, draw a line extending to the two sides of the circumference at the points P and Q. This will be a line perpendicular to the equinoctial ray, and it is called in mathematical figures the axis. From these same centres open the compasses to the ends of the diameters, and describe semicircles, one of which will be for summer and the other for winter.

6. Then, at the points at which the parallel lines cut the line called the horizon, the letter S is to be on the right and the letter V on the left, and from the extremity of the semicircle, at the point G, draw a line parallel to the axis, extending to the left hand semicircle at the point H. This parallel line is called the Logotomus. Then, centre the compasses at the point where the equinoctial ray cuts that line, at the letter D, and open them to the point where the summer ray cuts the circumference at the letter H. From the equinoctial centre, with a radius extending to the summer ray, describe the circumference of the circle of the months, which is called Menaeus. Thus we shall have the figure of the analemma

7. This having been drawn and completed, the scheme of hours is next to be drawn on the baseplates from the analemma, according to the winter lines, or those of summer, or the equinoxes, or the months, and thus many different kinds of dials may be laid down and drawn by this ingenious method. But the result of all these shapes and designs is in one respect the same: namely, the days of the equinoxes and of the winter and summer solstices are always divided into twelve equal parts. Omitting details, therefore, — not for fear of the trouble, but lest I should prove tiresome by writing too much,—I will state by whom the different classes and designs of dials have been invented. For I cannot invent new kinds myself at this late day, nor do I think that I ought to display the inventions of others as my own. Hence, I will mention those that have come down to us, and by whom they were invented.


1. The semicircular form, hollowed out of a square block, and cut under to correspond to the polar altitude, is said to have been invented by Berosus the Chaldean; the Scaphe or Hemisphere, by Aristarchus of Samos, as well as the disc on a plane surface; the Arachne, by the astronomer Eudoxus or, as some say, by Apollonius; the Plinthium or Lacunar, like the one placed in the Circus Flaminius, by Scopinas of Syracuse the προς τἁ ἱστοροὑμενα, by Parmenio; the προς παν κλιμα, by Theodosius and Andreas; the Pelecinum, by Patrocles; the Cone, by Dionysodorus; the Quiver, by Apollonius. The men whose names are written above, as well as many others, have invented and left us other kinds: as, for instance, the Conarachne, the Conical Plinthium, and the Antiborean. Many have also left us written directions for making dials of these kinds for travellers, which can be hung up. Whoever wishes to find their baseplates, can easily do so from the books of these writers, provided only he understands the figure of the analemma.

2. Methods of making water clocks have been investigated by the same writers, and first of all by Ctesibius the Alexandrian, who also discovered the natural pressure of the air and pneumatic principles. It is worthwhile for students to know how these discoveries came about. Ctesibius, born at Alexandria, was the son of a barber. Preeminent for natural ability and great industry, he is said to have amused himself with ingenious devices. For example, wishing to hang a mirror in his father's shop in such a way that, on being lowered and raised again, its weight should be raised by means of a concealed cord, he employed the following mechanical contrivance.

3. Under the roof-beam he fixed a wooden channel in which he arranged a block of pulleys. He carried the cord along the channel to the comer, where he set up some small piping. Into this a leaden ball, attached to the cord, was made to descend. As the weight fell into the narrow limits of the pipe, it naturally compressed the enclosed air, and, as its fall was rapid, it forced the mass of compressed air through the outlet into the open air, thus producing a distinct sound by the concussion.

4. Hence, Ctesibius, observing that sounds and tones were produced by the contact between the free; air and that which was forced from the pipe, made use of this principle in the construction of the first water organs. He also devised methods of raising water, automatic contrivances, and amusing things of many kinds, including among them the construction of water clocks. He began by making an orifice in a piece of gold, or by perforating a gem, because these substances are not worn by the action of water, and do not collect dirt so as to get stopped up.

 5. A regular flow of water through the orifice raises an inverted bowl, called by mechanicians the "cork" or "drum." To this are attached a rack and a revolving drum, both fitted with teeth at regular intervals. These teeth, acting upon one another, induce a measured revolution and movement. Other racks and other drums, similarly toothed and subject to the same motion, give rise by their revolution to various kinds of motions, by which figures are moved, cones revolve, pebbles or eggs fall, trumpets sound, and other incidental effects take place.

6. The hours are marked in these clocks on a column or a pilaster, and a figure emerging from the bottom points to them with a rod throughout the whole day. Their decrease or increase in length with the different days and months, must be adjusted by inserting or withdrawing wedges. The shutoffs for regulating the water are constructed as follows. Two cones are made, one solid and the other hollow, turned on a lathe so that one will go into the other and fit it perfectly. A rod is used to loosen or to bring them together, thus causing the water to flow rapidly or slowly into the vessels. According to these rules, and by this mechanism, water clocks may be constructed for use in winter.

7. But if it proves that the shortening or lengthening of the day is not in agreement with the insertion and removal of the wedges, because the wedges may very often cause errors, the following arrangement will have to be made. Let the hours be marked off transversely on the column from the analemma, and let the lines of the months also be marked upon the column. Then let the column be made to revolve, in such a way that, as it turns continuously towards the figure and the rod with which the emerging figure points to the hours, it may make the hours short or long according to the respective months.

8. There is also another kind of winter dial, called the Anaphoric and constructed in the following way. The hours, indicated by bronze rods in accordance with the figure of the analemma, radiate from a centre on the face. Circles are described upon it, marking the limits of the months. Behind these rods there is a drum, on which is drawn and painted the firmament with the circle of the signs. In drawing the figures of the twelve celestial signs, one is represented larger and the next smaller, proceeding from the centre. Into the back of the drum, in the middle, a revolving axis is inserted, and round that axis is wound a flexible bronze chain, at one end of which hangs the "cork" which is raised by the water, and at the other a counterpoise of sand, equal in weight to the "cork."

9. Hence, the sand sinks as the "cork" is raised by the water, and in staking turns the axis, and the axis the drum. The revolution of this drum causes sometimes a larger and sometimes a smaller portion of the circle of the signs to indicate, during the revolutions, the proper length of the hours corresponding to their seasons. For in every one of the signs there are as many holes as the corresponding month has days, and a boss, which seems to be holding the representation of the sun on a dial, designates the spaces for the hours. This, as it is carried from hole to hole, completes the circuit of a full month.

10. Hence, just as the sun during his passage through the constellations makes the days and hours longer or shorter, so the boss on a dial, moving from point to point in a direction contrary to that of the revolution of the drum in the middle, is carried day by day sometimes over wider and sometimes over narrower spaces, giving a representation of the hours and days within the limits of each month.

To manage the water so that it may flow regularly, we must proceed as follows.

11. Inside, behind the face of the dial, place a reservoir, and let the water run down into it through a pipe, and let it have a hole at the bottom. Fastened to it is a bronze drum with an opening through which the water flows into it from the reservoir. Enclosed in this drum there is a smaller one, the two being perfectly jointed together by tenon and socket, in such a way that the smaller drum revolves closely but easily in the larger, like a stopcock.

12. On the lip of the larger drum there are three hundred and sixty-five points, marked off at equal intervals. The rim of the smaller one has a tongue fixed on its circumference, with the tip directed towards those points; and also in this rim is a small opening, through which water runs into the drum and keeps the works going. The figures of the celestial signs being on the lip of the larger drum, and this drum being motionless, let the sign Cancer be drawn at the top, with Capricornus perpendicular to it at the bottom. Libra at the spectator's right, Aries at his left, and let the other signs be given places between them as they are seen in the heavens. 

 13. Hence, when the sun is in Capricornus, the tongue on the rim touches every day one of the points in Capricornus on the lip of the larger drum, and is perpendicular to the strong pressure of the running water. So the water is quickly driven through the opening in the rim to the inside of the vessel, which, receiving it and soon becoming full, shortens and diminishes the length of the days and hours. But when, owing to the daily revolution of the smaller drum, its tongue reaches the points in Aquarius, the opening will no longer be perpendicular, and the water must give up its vigorous flow and run in a slower stream. Thus, the less the velocity with which the vessel receives the water, the more the length of the days is increased.

14. Then the opening in the rim passes from point to point in Aquarius and Pisces, as though going upstairs, and when it reaches the end of the first eighth of Aries, the fall of the water is of medium strength, indicating the equinoctial hours. From Aries the opening passes, with the revolution of the drum, through Taurus and Gemini to the highest point at the end of the first eighth of Cancer, and when it reaches that point, the power diminishes, and hence, with the slower flow, its delay lengthens the days in the sign Cancer, producing the hours of the summer solstice. From Cancer it begins to decline, and during its return it passes through Leo and Virgo to the points at the end of the first eighth of Libra, gradually shortening and diminishing the length of the hours, until it comes to the points in Libra, where it makes the hours equinoctial once more.

15. Finally, the opening comes down more rapidly through Scorpio and Sagittarius, and on its return from its revolution to the end of the first eighth of Capricornus, the velocity of the stream renews once more the short hours of the winter solstice. 

The rules and forms of construction employed in designing dials have now been described as well as I could. It remains to give an account of machines and their principles. In order to make my treatise on architecture complete, I will begin to write on this subject in the following book.

Monday, 13 June 2016

Biography of Ramon Llull (1232-1316)

Image Source: Wikipedia
Source: "Dictionary of Scientific Biography, Volume 8" by Charles Coulston Gillispie (Ed.)
LULL, RAMON (b. Ciutat de Mallorques [now Palma de Mallorca], ca. 1232; d. Ciutat de Mallorques [?], January/March [?] 1316), polymathy.
A Catalan encyclopedist, Lull invented an "art of finding truth" which inspired Leibniz's dream of a universal algebra four centuries later. His contributions to science are understandable only when examined in their historical and theological context. The son of a Catalan nobleman of the same name who participated in the reconquest of Mallorca from the Moors, Lull was brought up with James the Conqueror's younger son (later crowned James 11 of Mallorca), whose seneschal he became. About six years after his marriage to Blanca Picany (1257) he was converted from a courtly to a religious way of life, following a series of visions of Christ crucified. He never took holy orders (although he may have become a Franciscan tertiary in 1295), but his subsequent career was dominated by three religious resolutions: to become a missionary and attain martyrdom, to establish colleges where missionaries would study oriental languages, and to provide them with "the best book[s] in the world against the errors of the infidel."{1}
Lull's preparations lasted a decade; his remaining forty years (from 1275, when he was summoned by Prince James to Montpellier, where he lectured on the early versions of his Art) were spent in writing, preaching, lecturing, and traveling (including missionary journeys to Tunis in 1292; Bougie, Algeria, in 1307; and Tunis late in 1315), and in attempts to secure support from numerous kings and four successive popes for his proposed colleges. During Lull's lifetime only James II of Mallorca established such a foundation (1276, the year of his accession); when he lost Mallorca to his elder brother, Peter 11I of Aragon, the college at Miramar apparently was abandoned (ca. 1292). In Lull's old age his proposals were finally approved by the Council of Vienne (1311-1312); and colleges for the study of Arabic, Hebrew, and Chaldean were founded in Rome, Bologna, Paris, Salamanca, and Oxford after Lull's death. Pious tradition has it that he died after being stoned by Muslims in Bougie (January 1316[?]), although his actual death is variously said to have occurred in Bougie, at sea, or in Mallorca; modern scholars doubt the historicity of his martyrdom. As for his third resolution, it led to the various versions of Lull's Art - and all his scientific contributions were by-products of this enterprise.
James the Conqueror's chief adviser, the Dominican Saint Ramon de Penyafort, dissuaded Lull from studying in Paris, where his age and lack of Latin would have told against him; he therefore studied informally in Mallorca (1265[?]-1273[?]). His thought was thus not structured at the formative stage by the Scholastic training which molded most other late medieval Christian thinkers; this fostered the development of his highly idiosyncratic system by leaving his mind open to numerous non-Scholastic sources. These included cabalism (then flourishing in learned Jewish circles in both Catalonia and Italy), earlier Christian writers discarded by Scholasticism (for instance, John Scotus Eriugena, whose ninth-century De divisione naturae influenced Lullian cosmological works, notably the Liber chaos, either directly or indirectly and hence also his Art), and probably also Arabic humoral medicine and astrology. The Augustinian Neoplatonism of the Victorines also proved important, partly because of its continuing prominence but mainly because its marked coincidences with both Islamic and cabalistic Neoplatonism favored the creation of a syncretistic system which was firmly grounded in doctrines equally acceptable to Christians, Jews, and Muslims.
This fusion occurred after the eight years Lull spent in Mallorca studying Latin, learning Arabic from a slave, reading all texts available to him in either tongue, and writing copiously. One of his earliest works was a compendium of the logic of al-Ghazali in Arabic (1270[?]); it has since been lost, although two later compendia with similar titles survive one in Latin, the other in Catalan mnemonic verse. In all, Lull wrote at least 292 works in Catalan, Arabic, or Latin over a period of forty-five years (1270-1315); most of them have been preserved, although no Arabic manuscripts have yet been traced and many Catalan and Latin works remain unpublished. His initial awkwardness in Latin, coupled with his desire that knowledge be made available to non-Latin-speaking sectors of society, made Lull the first person to mold Catalan into a literary medium. He used it not only in important mystical works, poetry, and allegorical novels (none of which concerns us here) but also to deal with every learned subject which engaged his attention: theology and philosophy; arithmetic, geometry, and astronomy (often mainly astrology), which, together with music, formed the quadrivium (the higher division of the seven liberal arts); grammar, rhetoric, and logic (the trivium); law; and medicine.
Thus, Lull created a fully developed learned vocabulary in Catalan almost a century before any other Romance vernacular became a viable scholarly medium. Almost all Lull's works in such nonliterary fields were connected in some way with his Art, because the "art of finding truth" which he developed to convert "the infidel" proved applicable to every branch of knowledge. Lull himself pioneered its application to all subjects studied in medieval universities-except for music-and also constructed one of the last great medieval encyclopedias, the Arbor scientiae (1295-1296), in accordance with its basic principles.
Yet the Art can be understood correctly only when viewed in the light of Lull's primary aim: to place Christian apologetics on a rational basis for use in disputations with Muslims, for whom arguments de auctoritate grounded on the Old Testament-widely used by Dominicans in disputations with the Jews carried no weight. The same purpose lay behind the Summa contra gentiles of Aquinas, written at the request of his fellow Dominican Penyafort, whose concern for the conversion of all non-Christians (but particularly those in James the Conqueror's dominions) thus inspired the two chief thirteenth-century attempts in this direction; the Summa contra gentiles was finished during the interval between Lull's discovery of his own calling and his interview with Penyafort. But whereas Aquinas distinguished categorically between what reason could prove and that which, while not contrary to reason, needed faith in revelation, Lull advanced what he called necessary reasons for accepting dogmas like the Trinity and the Incarnation. This gave his Art a rationalistic air that led to much subsequent criticism. Lull himself described his Art as lying between faith and logic, and his "necessary reasons" were not so much logical proofs as reasons of greater or lesser congruence which could not be denied without rejecting generally accepted principles. In this respect they were not appreciably more "rationalistic" than Aquinas's "proofs" that the truths of faith were not incompatible with reason. But the differences between the two apologetic systems are far more striking than their resemblances.
Lull regarded his Art as divinely inspired and hence infallible (although open to improvement in successive versions). Its first form, the Ars compendiosa inveniendi veritatem or Ars maior {2} (1273-1274[?]), was composed after a mystical "illumination" on Mount Randa, Mallorca, in which Lull saw that everything could be systematically related back to God by examining how Creation was structured by the active manifestation of the divine attributes-which he called Dignities and used as the absolute principles of his Art. Examining their manifestations involved using a set of relative principles; and both sets could be visualized in combinatory diagrams, known as Figures A and T. The original Figure A had sixteen Dignities, lettered BCDEFGHIKLMNOPQR; the original Figure T had five triads, only three of which (EFG + HIK + LMN) were strictly principles of relation, the others being sets of subjects (God + Creature + Operation, BCD) and possible judgments (Affirmation = Doubt Negation, OPQ). All early versions had a proliferation of supplementary visual aids, which always included diagrams showing the four elements, and-with the obvious exception of Figure T - most features of the system were grouped into sets of sixteen items, lettered like the Dignities.
This quaternary base seems to provide the key to the origins of the Art's combinatory aspect, apparently modeled on the methods used to calculate combinations of the sixteen elemental "grades" (four each for fire, air, water, and earth) in both astrology and humoral medicine. A major simplification in the Ars inventiva (ca. 1289) eliminated the elemental features, reduced the diagrams to four (unchanged thereafter), reduced Figure T to the nine actual relative principles (Difference + Concordance + Contradiction, Beginning + Middle + End, and Majority + Equality + Minority) and the sixteen original Dignities to the nine shown in Figure 1. In still later versions the symbolic letters BCDEFGHIK acquired up to six meanings that were ultimately set out in the gridlike "alphabet" of the Ars generalis ultima and its abridgment, the Ars brevis (both 1308), from which Figure 2 is reproduced. The traditional seven virtues and seven vices have been extended to sets of nine, to meet the requirements of the ternary system; the last two of ten quaestiones (a series connected with the ten Aristotelian categories) had to share the same compartment, since the set of fundamental questions could not be shortened and still be exhaustive.

The most distinctive characteristic of Lull's Art is clearly its combinatory nature, which led to both the use of complex semimechanical techniques that sometimes required figures with separately revolving concentric wheels-"volvelles," in bibliographical parlance (see Figure 3)-and to the symbolic notation of its alphabet. These features justify its classification among the forerunners of both modern symbolic logic and computer science, with its systematically exhaustive consideration of all possible combinations of the material under examination, reduced to a symbolic coding. Yet these techniques taken over from nontheological sources, however striking, remain ancillary, and should not obscure the theocentric basis of the Art. It relates everything to the exemplification of God's Dignities, thus starting out from both the monotheism common to Judaism, Christianity, and Islam and their common acceptance of a Neoplatonic exemplarist world picture, to argue its way up and down the traditional ladder of being on the basis of the analogies between its rungs-as becomes very obvious in Lull's De ascensu et descensu intellectus (1305). The lowest rung was that of the elements, and Lull probably thought that the "model" provided by the physical doctrines of his time constituted a valid "scientific" basis for arguments projected to higher levels. Since this physical basis would be accepted in the scientific field by savants of all three "revealed religions," he doubtless also hoped that the specifically Christian conclusions which he drew in the apologetic field would be equally acceptable. It even seems likely that what hit him with the force of a divine "illumination" on Mount Randa was his sudden recognition of such a possibility.
There is no evidence that Lull's Art ever converted anybody, but his application of the combinatory method to other disciplines (begun in the four Libri principiorum, ca. 1274--1275) was followed by numerous later Lullists; the Art's function as a means of unifying all knowledge into a single system remained viable throughout the Renaissance and well into the seventeenth century. As a system of logical inquiry (see Lull's Logica nova [1303] for the strictly logical implications, disentangled from other aspects), its method of proceeding from basic sets of preestablished concepts by the systematic exploration of their combinations-in connection with any question on any conceivable subject-can be succinctly stated in terms taken from the Dissertatio de arte combinatoria (1666) of Leibniz, which was inspired by the Lullian Art: "A proposition is made up of subject and predicate; hence all propositions are combinations. Hence the logic of inventing [discovering] propositions involves solving this problem: 1. given a subject, [finding] the predicates; 2. given a predicate, finding the subjects [to which it may] apply, whether by way of affirmation or negation." {3}
Recent research has concentrated on the clarification of Lull's ideas, the identification of their sources, and the nature of their influence on later thinkers - especially Nicholas of Cusa and Giordano Bruno. Major advances in all these fields have taken place since the 1950's, but much more research is still required. The specific origins of Lull's doctrines regarding the elements, whose importance has been fully recognized only since 1954 (see Yates), are particularly significant. A proper exploration of the antecedents of his Opera medica is a prerequisite for establishing Lull's final place in the history of Western science. In this connection it must be mentioned that although Lull himself was opposed to alchemy (but not to astrology, a "science" he sought to improve in the Tractatus novas de astronomia [1297]), his methods had obvious applications in the alchemical field and they were so applied in a host of pseudo-Lullian alchemical works, most of them composed more than fifty years after his death. These works explain the traditional (but false) "scientific" view which made him "Lull the Alchemist."

{1} Vida coetanea. The Latin (dictated by Lull [?], probably 1311) says "book," which doubtlessly agreed with Lull's original resolve; the plural, in the fourteenth-century Catalan text (modernized in Obres essencials, see I, 36), would better fit the series of "improved" versions of the Art itself, which first took shape almost ten years after Lull's conversion.
{2} References to an Ars magna in later centuries are usually either to the definitive Ars magna generalis ultima (1308) or to Lull's system in general. The alternative title of the first version recalls Roger Bacon's Opus maius (1267); the connections between Lull and Bacon have yet to be investigated, but many resemblances may well be due to common Arabic sources.
{3} "Propositio componitur ex subiecto et praedicato, omnes igitur propositiones sunt combinationes. Logicae igitur inventivae propositionum est hoc problenia solvere: 1. Dato subiecto praedicata. 2. Dato praedicato subiecta invenire, uiraque cum affirmative, turn negative" (G. W. Leibniz, op. cit. [in text], no. 55, in Sauntliche Schriften and Briefen, 2nd ser., I [Darmstadt, 1926], 192).