علم الآلات الحربيةGeometry Surveying Mechanics and Engineering

The Science of Military Machines

It is the science by which one learns how to construct military machines, such as the catapult and the like.

It is a branch of geometry.

Its benefit is evident, for it is of great importance in repelling enemies and protecting cities.

This science is one of the pillars of religion, since the conduct of jihaad depends upon it. The two sons of Moosaa ibn Shaakir wrote a useful book on this science; so it is stated in the Key to Happiness and the City of the Sciences.

The science of archery and firearms should be added to this science, with the reminder that sciences of this kind are of two sorts:

The science of designing and manufacturing them.

And the science of using them; books have been written on it.

The Science of Pneumatic Machines

It is the science by which one learns how to construct machines based on the impossibility of a vacuum and related principles, such as the justice cup and the lover's cup.

The first is a vessel which, when filled to a certain measure, retains the drink; if even a little more is added, the water pours out and the vessel empties itself so that not a drop remains.

The second has a certain capacity: if water is poured into it in the smaller specified amount, it remains; if it is filled, it also remains; but if the amount lies between the two measures, the vessel empties itself. All this is due to the impossibility of a vacuum.

This science, insofar as it concerns a specified measure of a vessel, is a branch of geometry; insofar as it is based on the impossibility of a vacuum, it is a branch of natural philosophy. The rotation of clocks belongs to the same category. It is called the science of pneumatic machines because the soul finds relief and exercise in the marvels of these machines.

The best-known books in this field are the Tricks of the Sons of Moosaa ibn Shaakir, a concise book by Qaylan, and an extensive book by Badee' al-Jazaree. So said Aboo al-Khayr.

The Science of Water-Sourcing

It is the science by which one learns how to extract hidden waters from the earth and bring them to the surface. Its benefit is evident: the revival and cultivation of lands. It is related from some scholars that he said: “If God's servants knew that God is pleased when His land is revived, no uninhabited place would remain upon the face of the earth.”

Al-Karkhee wrote a concise book on it, and the Nabaean Agriculture contains important material on this science. This concludes what is found in the City of the Sciences and the Key to Happiness; the distinguished Aboo al-Khayr also listed it among the branches of geometry.

The Science of Planar Projection of the Sphere

It is the science by which one learns how to produce radial instruments; so it is stated in the Glossary of Technical Terms of the Sciences.

The Disclosure of Hidden Doctrines says: “It is the manner of transferring the sphere to a plane while preserving the lines and circles drawn on the sphere, and the manner of transferring those circles from a circle to a line. Conceiving this science is extremely difficult, almost like violating the customary order of nature; but carrying it out by hand is often undertaken by people and is not as difficult as conceiving it.” This is what Aboo al-Khayr mentioned.

He made it a branch of astronomy, which is itself a branch of geometry. The claim that it is difficult to conceive is not unrestricted; it is difficult only for one who has not practiced geometry.

Its benefit lies in exercising oneself through knowledge and construction of these instruments, extracting them from mental principles corresponding to external configurations, and using them to derive astronomical conclusions.

Books composed on it include Ptolemy's Planar Projection of the Sphere; al-Farghaanee's Complete Work; al-Beerootee's Comprehensive Treatment; Taqee al-Deen's Constitution and Preference in the Principles of Projection; and al-Marraakushee's Instruments of Calendar Computation—may God have mercy on them both.

The Science of Moving Heavy Loads

It is the science that investigates how to construct machines that move heavy objects with slight force.

Its benefit is evident even to ordinary people. Ideen demonstrated in his book on this science how one hundred thousand pounds could be moved with a force of five hundred, a feat that limited minds find hard to accept.

It is a branch of geometry. The Imaam demonstrated some of its problems at the end of the جامع العلوم. The author of the Key to Happiness, and likewise the author of the City of the Sciences, mentioned no book on this art; but in our time many books on it have appeared in the language of the Franks, who have great expertise in it. In our own age they have produced devices that move enormous weights and loads over very great and difficult distances in short and easy periods, leaving minds bewildered and pens unable to record them. Among them is the steam carriage, which travels the distance of a month in a day and a night.

The Science of Mathematics

The mathematical sciences are among the divisions of theoretical wisdom.

It is a science that investigates material things which can be abstracted from matter in thought. It is called this because it was customary for philosophers to exercise themselves in it at the beginning of their instruction of children; hence it is also called an educational science. It is the intermediate science, standing between what does not require matter and what absolutely does require it, since it requires matter in one respect and does not require it in another.

It has principles, each of which has branches. Its four principles are geometry, astronomy, arithmetic, and music, because its subject is quantity. Quantity is either continuous or discrete.

The continuous is either moving or stationary: the moving is astronomy, and the stationary is geometry.

The discrete is either possessed of a compositional ratio or not. The former is music; the latter is arithmetic, whose branches are six:

First: addition and subtraction.

Second: algebra and restoration.

Third: mensuration.

Fourth: the moving of heavy loads.

Fifth: astronomical tables and calendars.

Sixth: the science of organ-making, namely, the construction of unusual instruments.

The author of the Glossary of Technical Terms of the Sciences says: “Mathematics is the science of the states of things that, in external existence but not in intellection, require matter, such as squaring, cubing, circular construction, spherical and conical forms, number, and its properties. These are things that require matter for their existence, but not for their definitions. It is also called intermediate wisdom.”

The ancient philosophers differed over whether natural philosophy or mathematics deserved greater honor and excellence. Each inclined toward one side with arguments recorded among them. The truth is that judging one categorically superior to the other is unsound; each is superior to the other in one respect.

Natural philosophy is superior to mathematics because its subject is a natural body, which is a substance, whereas the subject of mathematics is quantity, which is an accident; substances are nobler than accidents. Natural philosophy also, for the most part, gives the “why,” while mathematics gives the “that”; and giving the why is superior. It also includes the science of the soul, which is the mother of wisdom and the source of the virtues.

Mathematics is superior to natural philosophy because imaginary and imaginative states are infinitely divisible and therefore have no stopping point, whereas mathematics is bounded by limits. Mathematical things are also purer, subtler, more pleasurable, and more complete than bodily things that cloud the mind. Confusion and error are less frequent in its numerical and geometrical demonstrations than in natural philosophy—and even in divine philosophy. For this reason it was said that apprehending divine and natural matters is, in their likeness and probability, not a matter of certainty. This is stated in al-Sadr; here is the substance of his discussion.

The Science of Building Structures

It is a science by which one learns the states of building arrangements, how to make them sound, and the means of making them beautiful: constructing secure fortresses, arranging splendid houses, erecting strong bridges, and the like. It also treats the manner of digging rivers, straightening channels, sealing breaches,[Edition footnote 1] bringing forth water, conveying it from lowlands to highlands, and other such matters. Its benefit for building cities, houses, and citadels, and for agriculture, is plainly immense. Ibn al-Haytham wrote a book on it, as did al-Karkhee. The Christian rulers of India—the British—also have great expertise in this science.

[Edition footnote 1] The verb means “to burst forth”: a river bursts open and water gushes out, as from a spring. The place where this occurs is called a breach, and its plural is breaches. (Qaamoos.)

The Science of Centers of Gravity

Aboo al-Khayr says in the Key to Happiness: “It is the science by which one learns how to determine the center of gravity of a carried body. The center of gravity means a point in the body at which it balances with respect to its support.”

Its benefit is knowing how to balance large bodies with smaller ones by means of the intervening distance. This concludes the passage.

Aboo Sahl al-Koohee wrote a book on it, though he was lax in the premises of his demonstrations; Ibn al-Haytham also wrote a useful book on it.

The Science of Burning Mirrors

Aboo al-Khayr says: “It is the science by which one learns the states of bent, reflected, and refracted rays; their locations, angles, and points of return; and how to make burning mirrors by reflecting the sun's rays from them, and how to set and align them. Its benefit is considerable in the besieging of cities and fortresses.” This is likewise stated in the Glossary of Technical Terms of the Sciences.

The ancients used to make mirrors from flat surfaces; some made them from the concave surface of a sphere, until Anthemius appeared and demonstrated that if their surfaces were concave according to the appropriate section, they would be extraordinarily powerful and incendiary. Ibn al-Haytham's book on burning mirrors follows this view, as stated in the City of the Sciences.

The Science of Surveying

This is how it is given in the Disclosure of Hidden Doctrines. I say: it is a branch of geometry, and it is an art needed for surveying land.

It means determining the measure of a known tract of land in relation to a span, cubit, or other unit, or determining the ratio of one tract to another when measured by such a unit. It is needed for assessing the land tax on farms and cultivated fields,[Edition footnote 2] for orchards, and for dividing walls and lands among partners or heirs, and similar matters. People have written many excellent works on it. God grants success in what is correct. This concludes what is found in Ibn Khaldoon.

[Edition footnote 2] Arable plots.

The City of the Sciences defines it as follows: “It is the science by which one learns the measures of lines, surfaces, and bodies, using the line, square, and cube as measures.”

Its benefit is great in matters of taxation, the division of lands, the estimation of dwellings, and other matters.

Among its concise books is a book by Ibn Mahallee al-Mawsilee. Among the intermediate works are a book by Ibn al-Mukhtaar and a book by Schmid. This concludes the passage.

This science is widespread among people today, and the Christians who rule India have the greatest knowledge of it. God Most High knows best what is correct.

The Science of Optics

It is a branch of geometry. It explains the causes of error in visual perception by showing how they occur, on the premise that sight perceives by means of a cone of rays whose apex is at the observer and whose base is the visible object. Thus one often sees what is near as large and what is far as small; small images beneath water and behind transparent bodies as large; a falling raindrop as a straight line; and a flame as a circle, along with similar phenomena. This science explains the causes and modes of these errors through geometrical demonstrations. It also explains the changing apparent position of the moon according to differences in latitude, upon which knowledge of crescent visibility and the occurrence of eclipses depends, and many similar matters.

Many Greeks composed works in this field. Among Muslims, the most famous author was Ibn al-Haytham, though others also wrote on it. It belongs to mathematics and its branches, as Ibn Khaldoon stated.

The City of the Sciences explains optics as follows: “It is the science by which one learns the states of visible things in their quantity and quality, with respect to their nearness to or distance from the observer, the differences in their forms and positions, and the thickness and thinness of what lies between the observer and the visible objects. It investigates the causes of these matters. Its benefit is knowledge of the states of vision, the variation among visible objects, and the causes of sensory errors occurring in them. This science is also used in surveying distant bodies and in burning mirrors.”

Among its concise books is Euclid's work; among the intermediate books is the book by 'Alee ibn 'Eesaa the vizier; and among the extensive books is Ibn al-Haytham's work. The Glossary of Technical Terms of the Sciences gives a similar account in abbreviated form.

The Science of Geometry

It is the science of laws by which one learns the properties that attach to quantity insofar as it is quantity. The City of the Sciences says:

“It is the science by which one learns the states of magnitudes, their concomitants, the arrangements of some in relation to others, the ratios among them, the properties of their shapes, the methods for producing what can be produced through them, and the extraction of what needs to be extracted by certain demonstrations.

“Its subject is absolute magnitudes—namely, the line, plane, and mathematical solid—and their concomitants, such as the angle, point, and figure.

“Its benefit is acquaintance with these states as they occur in existing things, and the acquisition of sharpness and penetration of mind. It gives thought rigorous exercise, since it is agreed that the sciences with the strongest demonstrations are the geometrical sciences.

“Among its benefits is treatment for compounded ignorance, because these are certain sciences in which imagination has no entry. The mind thereby becomes accustomed to bringing imagination under control; compounded ignorance is nothing other than imagination's domination over the intellect.

“Its books are numerous. The most famous and soundest is al-Toosee's revision of Euclid's Elements. The shortest and best is al-Abharee's commentary on the Forms of the Foundations, together with the commentary of Qaadheezaada al-Roomee. Ibn Seenaa included an adequate account of it in the Book of Healing, and the distinguished author also recorded in his books enough of the truths of this art.” This concludes the passage.

The word geometry is an Arabized form of andaazah, and the reason for the name is clear.

The sciences branching from it are ten. This is because one either investigates the actual production of what is demonstrated in the universal principles, or one does not.

The latter is either concerned first with what is observed or not. The second is the science of building structures. If the science concerned with what is observed is restricted to the reflection of rays, it is the science of burning mirrors; otherwise it is the science of optics.

As for the former—the investigation of actually producing the sought object from universal principles—it is either concerned with estimating it or not. If the former is restricted to transfer, it is the science of centers of gravity; otherwise it is surveying.

The latter is either concerned with producing instruments or not. The second is the science of water-sourcing. Instruments are either measuring instruments or not.

Measuring instruments are either heavy—this is the science of moving heavy loads—or temporal—this is the science of water clocks.

Those that are not measuring instruments are either military or not. The second is the science of pneumatic machines.

The first is the science of military machines. We have mentioned these sciences in this book in alphabetical order; consult their entries.

Ibn Khaldoon, may God have mercy on him, said: “This science investigates magnitudes, whether continuous—such as line, plane, and solid—or discrete—such as numbers—and the essential accidents that occur to them.

“For example: every triangle has angles equal to two right angles.

“Every two parallel lines do not meet, even if extended without limit.

“When two lines intersect, their vertically opposite angles are equal.

“The product of the first and third of four proportionate magnitudes equals the product of the second and fourth, and similar propositions.”

The book translated from the Greeks in this art is Euclid's book, called the Elements and also the Book of the Principles. It is the simplest work composed for students in this field.

The first translation of a Greek book in this discipline among Muslims was made in the days of Aboo Ja'far al-Mansoor. Its copies differ according to the translators: some were by Hunayn ibn Ishaaq, Thaabit ibn Qurrah, and Yoosuf ibn al-Hajjaaj. It contains fifteen books: four on planes; one on proportional magnitudes; another on the ratios of planes to one another; three on number; the tenth on rational quantities and powers of rational quantities—meaning roots; and five on solid figures.

Many abridgements were made of it, as Ibn Seenaa did in the mathematical section of the Healing, to which he devoted a separate volume, and as Ibn al-Salt did in the Book of Economy, among others.

Others wrote many commentaries on it. It is the foundation of the geometrical sciences in the broadest sense.

Know that geometry gives its practitioner illumination of mind and soundness of thought, because all its demonstrations are plainly ordered and clearly arranged. Error scarcely enters its syllogisms because of their order and regularity. By practicing it, thought is distanced from error, and its practitioner develops an intellect along that path. They say that it was written over Plato's door: “Let no one who is not a geometer enter our house.”

Our teachers, may God Most High have mercy on them, used to say: “Practicing geometry for the mind is like soap for a garment: it washes away its dirt and purifies it of filth and stains.” This is due to the order and regularity we have mentioned.

Among the branches of this art is the geometry devoted to spherical and conical figures.

As for spherical figures, the Greeks have two books by Theodosius and Menelaus on their surfaces and sections. Theodosius's book precedes Menelaus's in instruction, since many of its demonstrations depend upon it. Both are indispensable to anyone who wishes to enter astronomy, because its demonstrations depend upon them. Discussion of astronomy is entirely discussion of the celestial spheres and the sections and circles that occur in them through the causes of motion, as we shall mention. Knowledge of the laws governing spherical figures, their surfaces, and their sections may therefore be necessary.

As for conics, they are also a branch of geometry. It investigates the figures and sections that occur in conical bodies and demonstrates, through geometrical proofs dependent on the first instruction, the properties that occur in them.

Their benefit appears in practical crafts whose materials are bodies, such as carpentry and building; in making unusual statues and rare structures; and in devising means of moving heavy loads and transporting structures by means of frames, winches, and similar devices.

Some authors devoted a book to practical contrivances in this field. It contains every marvel of unusual crafts and ingenious devices, though its demonstrations may be difficult for minds to grasp because of their geometrical complexity. It is in people's hands and is attributed to the Sons of Shaakir. God Most High knows best.