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THE doctrines of mathematics are so necessary to and have such an affinity with magic, that they who profess it without them are quite out of the way, and labour in vain, and shall in no wise obtain their desired effect. For whatsoever things are, and are done in these inferior natural virtues, are all done and governed by number, weight, measure, harmony, motion, and light: and all things which we see in these inferiors have root and foundation in them; yet, nevertheless, without natural virtues of mathematical doctrines, only works like to naturals can be produced: as Plato saith--a thing not partaking of truth or divinity, but certain images akin to them (as bodies going, or speaking, which yet want the animal faculty), such as were those which, amongst the ancients, were called Dedalus's images, and αυτοματα, of which Aristotle makes mention, viz. the three-footed images of Vulcan and Dedalus moving themselves; which, Homer saith, came out of their own accord to the exercise; and which, we read, moved themselves at the feast of Hiarba, the philosophical exerciser. So there are made glasses (some concave, others of the form of a column) making the representation of things in the air seem like shadows at a distance; of which sort Apollonius and Vitellius, in their books, "De Prospectiva," and "Speculis," taught the making and the use. And we read that Magnus Pompeius brought a certain glass, amongst the spoils from the East, to Rome, in which were seen armies of armed men. And there are made certain transparent glasses, which (being dipped in some certain juices of herbs, and irradiated with an artificial light) fill the whole air round about with visions. And we know how to make reciprocal glasses, in which the sun shining, all things which were illustrated by the rays thereof are apparently seen many miles off. Hence a magician (expert in natural philosophy and mathematics, and knowing the middle sciences, consisting of both these, viz. arithmetic, music, geometry, optics, astronomy, and such sciences that are of

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weights, measures, proportions, articles, and joints; knowing, also, mechanical arts resulting from these) may, without any wonder, if he excel other men in the art and wit, do many wonderful things, which men may much admire. There are some relics now extant of the antients, viz. Hercules and Alexander's pillars; the gate of Caspia, made of brass, and shut with iron beams, that it could by no art be broken; and the pyramids of Julius Cæsar, erected at Rome, near the hill Vaticanus; and mountains built by art in the middle of the sea; and towers, and heaps of stones, such as I have seen in England, put together by incredible art. But the vulgar seeing any wonderful sight, impute it to the Devil as his work; or think that a miracle which, indeed, is a work of natural or mathematical philosophy. But here it is convenient that you know, that, as by natural virtues we collect natural virtues, so by abstracted, mathematical, and celestial, we receive celestial virtues; as motion, sense, life, speech, soothsaying, and divination even in matter less disposed, as that which is not made by nature, but only by art. And so images that speak, and foretel things to come, are said to be made: as William of Paris relates of a brazen-head, made under the rising of Saturn, which, they say, spake with a man's voice. But he that will chuse a disposed matter, and most fit to receive, and a most powerful agent, shall undoubtedly produce more powerful effects. For it is a general opinion of the Pythagoreans, that, as mathematical are more formal than natural, so also they are more efficacious; as they have less dependance in their being, so also in their operation. But amongst all mathematical things, numbers, as they have more of form in them, so also are more efficacious, as well to affect what is good as what is bad. All things, which were first made by the nature of things in its first age, seem to be formed by the proportion of numbers; for this was the principal pattern in the mind of the Creator. Hence is borrowed the number of the elements--hence the courses of times--hence the motion of the stars, and the revolution of the heavens, and the state of all things subsist by the uniting together of numbers. Numbers, therefore, are endowed with great and sublime virtues. For it is no wonder, seeing there arc so many occult virtues in natural things, although of manifest operations, that there should be in numbers

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much greater and more occult, and also more wonderful and efficacious; for as much as they are more formal, more perfect, and naturally in the celestials, not mixed with separated substances; and, lastly, having the greatest and most simple commixion with the ideas in the mind of God, from which they receive their proper and most efficacious virtues; wherefore they also are of most force, and conduce most to the obtaining of spiritual and divine gifts--as, in natural things, elementary qualities are powerful in the transmuting of any elementary thing. Again, all things that are, and are made, subsist by and receive their virtue from numbers:--for time consists of numbers--and all motion and action, and all things which are subject to time and motion. Harmony, also, and voices have their power by and consist of numbers and their proportions; and the proportion arising from numbers do, by lines and points, make characters and figures; and these are proper to magical operations--the middle, which is betwixt both, being appropriated by declining to the extremes, as in the use of letters. And lastly, all species of natural things, and of those which are above Nature, are joined together by certain numbers; which Pythagoras seeing, says, that number is that by which all things subsist, and distributes each virtue to each number. And Proclus says, number hath always a being: yet there is one in voice--another in proportion of them--another in the soul and reason--and another in divine things. But Themistius, Boetius, and Averrois (the Babylonian), together with Plato, do so extol numbers, that they think no man can be a true philosopher without them. By them there is a way made for the searching out and understanding of all things knowable;--by them the next access to natural prophecying is had--and the Abbot Joachim proceeded no other way in his prophecies, but by formal numbers.

Next: Chapter XV: The Great Virtues Of Numbers, As Well In Natural Things As In Supernatural