The following examples will shew in what manner the proportion between the measures of any two given countries may be ascertained.
Examples.
Let it be required to reduce 100 archines of Russia into varas of Spain.
The archine measuring 336 English lines, and the vara 395,25, according to the table prefixed, I state the following equation:
Reduce 100 varas into archines.
EnglishLongMeasure.
inch3palm93span1241¹⁄₄foot18621¹⁄₂cubit3612432yard451553³⁄₄2¹⁄₂1¹⁄₄ell60206²⁄₃53¹⁄₂1³⁄₄1¹⁄₄pacefath-om722486421³⁄₄1¹⁄₅198662216¹⁄₂115¹⁄₂43¹⁄₁₀2³⁄₄polefur-long792026408806604402201761321104063360211207040528035201760140810568803208mile
JewishLong or ItineraryMeasure.
RomanlongMeasure,reduced toEnglish.
Eng.paces.ft.dec.inchdigitus transversus000.725¹⁄₄1³⁄₄uncia000.96743palmus minor002.90116124pes0011.604201551¹⁄₄palmipes012.505241861¹⁄₂1¹⁄₅cubitus015.4064030102¹⁄₂21²⁄₃gradus015.01806020543¹⁄₂2passus0410.021000075002500625500416²⁄₃250215stadium12044.5800006000020000500040003333¹⁄₃200010008milliare96700
English square or superficialMeasures, are raised from the yard of 36 inches multiplied into itself; and this producing 1296 square inches in the square yard, the divisions of this are square feet and inches, and the multipliers, poles, roods, and acres.
EnglishsquareMeasure.
Long Measure.
Square Measure.
Solid, or Cubic Measure.
Dry Measure.
Avoirdupois Weight.
French squareMeasures, are regulated by 12 square lines in the inch square, 12 inches in the foot, 22 feet in the perch, and 100 perches in the arpent or acre.
French liquidMeasures. At Paris, and in a great part of the kingdom, the smallest measure is the possou, which contains six cubic inches; 2 possous make the demi-septier; 2 demi-septiers the chopine; 2 chopines a pint; 2 pints a quart or pot; 4 quarts the gallon, or septier of estimation; 36 septiers the muid; which is subdivided into 2 demi-muids, 4 quarter muids, and 8 half quarter muids. The queue in Orleans, Blois, &c. contains a Paris muid and a half. The tun used at Bayonne and Bourdeaux, consists of 4 bariques, and equal to 3 Paris muids; at Orleans to 2: so that the first tun contains 864 pint, and the second 576. The demi-queue in Champagne, 96 quarts; the pipe in Anjou and Poictou, 2 bussards, equal to a demi-queues of Orleans, &c. or a muid and a half of Paris. The millerolle used in Provence, contains 66 Paris pints; and the poincon at Nantz, in Touraine, and the Blessois, equal to half the Orleans tun. The poincou at Paris is the same with the demi-queue.
French Weights and Measures.
The toise is commonly used in France for military purposes, and is divided into 6 feet: each foot 12 inches; each inch 12 lines; each line 12 points. The pace is usually reckoned at 2¹⁄₂ feet.
Poids de Marc, ou de Paris.
The French have lately formed an entire new system of weights and measures: the following short account of them, and their proportion to the old weights and measures of France, and those of English standard, is extracted fromNicholson’s Natural Philosophy.
Proportions of the measures of each species to its principal measure or unity.First part of the name which indicates the proportion to the principal measure or unity.PRINCIPAL MEASURES, OR UNITIES.Length.Capacity.Weight.Agrarian.For Firewood.10,000Myria-Metre.Litre.Gramme.Are.Stere.1,000Kilo100Hecto10Deca0——0.1Deci0.01Centi0.001MilliProportion of the principal measures between themselves, and the length of the Meridian.10,000,000th part of the dist. from the Pole to the Equator.A Decimetre cube.Weight of a centimetre cube of distilled Water.100 square metres.One cubic metre.Value of the principal measures in the ancient French measures.3 feet 11 lines and ¹⁄₂ nearly.1 pint and ¹⁄₂₀ or 1 litron and ¹⁄₄ nearly.18 grains and 841,000 parts.Two square perches des eaux et foret.1 demi voie or ¹⁄₄ of a cord des eaux et foret.Value in English measures.Inches 39·383.61.083 inch, which is more than the wine and less than the beer quart.22·966 grains.11·968 square yards.
By the new metrical system of the French, the geometrical circle used in astronomical, geographical, and topographical calculations, is divided instead of 360, into 400 equal parts, which are calledgrades: each grade is divided into 100 equal parts which are calledminutesofgrades: and each minute into 100seconds, ofgrades. The proportion of the new to the old degree is 0.9; and the next proportion or minute is 54′ of the old division; and the new second is 32″.4 of the ancient.
Reduction of the old French Weights and measures to English; and the contrary.
GermanMeasures. The Rhinland rood is the measure commonly used in Germany and Holland, and in most of the northern states, for all military purposes.
It is divided into 12 feet. The Rhinland rood is sometimes divided into tenths, or decimal feet, and the pace is made equal to 2 decimal feet, or ²⁄₁₀ of a rood.
Proportions between the English Weights and Measures, and those of the principal Places in Europe.
CubicalMeasures, or measures of capacity for liquors. English liquid measures were originally raised from troy weight, it being ordained that pounds troy of wheat, gathered from the middle of the ear, and well dried, should weigh a gallon of wine measure; yet a new weight, viz. the avoirdupois weight, had been introduced, to which a second standard gallon was adjusted, exceeding the former in the proportion of the avoirdupois weight to the troy weight. From this latter standard were raised two measures, the one for ale, the other for beer.
The sealed gallon at Guildhall, London, which is the English standard for wine, spirits, oil, &c. is supposed to contain 231 cubic inches; yet by actual experiment made in 1688, before the lord mayor and commissioners of excise, it only contains 224 cubic inches. It was however agreed to continue the common supposed contents of 231: hence, as 12 : 231 ∷ 14¹²⁄₂₀ : 281¹⁄₂ the cubic inches in an ale gallon; but in effect, the ale quart contains 70¹⁄₂ cubic inches; on which principles the ale and beer gallon will be 282 cubic inches.
DryMeasure, is different from both the ale and wine measure, being nearly a mean between both.
According to a British act of parliament, passed in 1697, every round bushel with a plain and even bottom, being 18¹⁄₂ inches throughout, and eight inches deep, is to be accounted a legal Winchester bushel, according to the standard in the exchequer; consequently a corn gallon will contain 268.8 inches, as in the following table.
Winchester Measure.
Table Cloth Measure.
Measureof wood for firing, is the cord, being four feet high, as many broad, and the length of the wood is as by law established, it is divided into two half cords.
Measurefor horses, is the hand, which by statute contains 4 inches.
PowderMeasures, made of copper, holding from an ounce to 12 pounds, are very convenient in a siege, when guns or mortars are to be loaded with loose powder, especially in ricochet-firing, &c.
The French recommend measures that are made of block tin, such as are used for measuring out salt, viz. 1 ounce, 2, 3, 4, 8, which make the half pound; and lastly, of 16, which make the pound. These quantities answer every sort of ordnance.
Diameters and Heights of Cylindric Powder Measures, holding from 1 to 15 Ounces.
Diameters and Heights of Cylindric Powder Measures, holding from 1 to 15 Pounds.
The above are in inches and decimals.
Measure-angle, a brass instrument to measure angles, either saliant or rentrant, for exactly ascertaining the number of degrees and minutes, to delineate them on paper.
MEASURING,-MENSURATION,
inmilitary mathematics, the assuming any certain quantity, and expressing the proportion of other similar quantities to the same; or the determining, by a certain known measure, the precise extent, quantity, or capacity of any thing.
Measuring, ingeneral, constitutes the practical part of geometry; and from the various subjects which it embraces, it acquires various names, and constitutes various arts, viz.
Longimetry,Altimetry,Levelling,Geodesia, orSurveying,Stereometry,Superficies, andSolids, &c. which see.
Measuring.SeeChain.
MECHANICS, a mixed mathematical science, which considers motion and moving powers, their nature and laws, with the effects thereof, in machines, &c. The word is derived from the Greek. That part which considers motion arising from gravity, is sometimes called statics, in contradistinction from that part which considers the mechanical powers and their application, properly called mechanics: it is, in fine, the geometry of motion.
Mechanics.The whole momentum or quantity of force of a moving body, is the result of the quantity of matter, multiplied by the velocity with which it is moved; and when the product arising from the multiplication of the particular quantities of matter in any two bodies, by their respective velocities are equal, their momentum will be so too. Upon this easy principle depends the whole of mechanics; and it holds universally true, that when two bodies are suspended on any machine, so as to act contrary to each other; if the machine be put in motion, and the perpendicular ascent of one body multiplied into its weight, be equal to the perpendicular descent of the other, multiplied into its weight, those bodies, how unequal soever in their weights, will balance each other in all situations: for, as the whole ascent of the one is performed in the same time as the whole descent of the other, their respective velocities must be as the spaces they move through; and the excess of weight in one is compensated by the excess of velocity in the other. Upon this principle it is easy to compute the power of any engine, either simple or compound; for it is only finding how much swifter the power moves than the weight does, (i. e.how much further in the same time,) and just so much is the power increased by the help of the engine.
The simple machines usually called mechanic powers, are six in number,viz.thelever, thewheel and axle, thepulley, theinclined plane, thewedge, and thescrew.
There are four kinds oflevers: 1st, where the prop is placed between the weight and the power. 2d, where theprop is at one end of the lever, the power at the other, and the weight between them. 3d, where the prop is at one end, the weight at the other, and the power applied between them. 4th, the bended lever, which differs from the first in form, but not in property.
In the first and 2d kind, the advantage gained by the lever, is as the distance of the power from the prop, to the distance of the weight from the prop. In the 3d kind, that there may be a balance between the power and the weight, the intensity of the power must exceed the intensity of the weight, just as much as the distance of the weight from the prop exceeds the distance of the power from the prop. As this kind of lever is disadvantageous to the moving power, it is seldom used.
Wheel and axle.Here the velocity of the power is to the velocity of the weight, as the circumference of the wheel is to the circumference of the axle.
Pulley.A single pulley, that only turns on its axis, and does not move out of its place, serves only to change the direction of the power, but gives no mechanical advantage. The advantage gained in this machine, is always as twice the number ofmoveablepullies; without taking any notice of thefixedpullies necessary to compose the system of pullies.
Inclined plane.The advantage gained by the inclined plane, is as great as its length exceeds its perpendicular height. The force wherewith a rolling body descends upon an inclined plane, is to the force of its absolute gravity, as the height of the plane is to its length.
Wedge.This may be considered as two equally inclined planes, joined together at their bases. When the wood doesnotcleave at any distance before the wedge, there will be an equilibrium between the power impelling the wedge, and the resistance of the wood acting against its two sides; when the power is to the resistance, as half the thickness of the wedge at the back, is to the length of either of its sides; because the resistance then acts perpendicular to the sides of the wedge: but when the resistance on both sides acts parallel to the back, the power that balances the resistance on both sides will be, as the length of the whole back of the wedge is to double its perpendicular height. When the wood cleaves at any distance before the wedge, (as it generally does) the power impelling the wedge will be to the resistance of the wood, as half the length of the back is to the length of either of the sides of the cleft, estimated from the top, or acting part of the wedge.
Screw.Here the advantage gained is as much as the circumference of a circle described by the handle of the winch, exceeds the interval or distance between the spirals of the screw.
There are few compound engines, but what, on account of the friction of parts against one another, will require a third part more power to work them when loaded, than what is required to constitute a balance between the power and the weight.
MECHANICAL, something relating to mechanics.
Mechanicalphilosophy, that which explains the phenomena of nature, and the operations of corporeal things, on the principles of mechanics; namely, the motion, gravity, figure, arrangement, &c. of the parts which compose natural bodies.
Mechanicalpowers. When two heavy bodies or weights are made by any contrivance to act against each other, so as mutually to prevent each other, from being put into motion by gravity, they are said to be in equilibrio. The same expression is used with respect to other forces, which mutually prevent each other from producing motion.
Any force may be compared with gravity, considered as a standard. Weight is the action of gravity on a given mass. Whatever therefore is proved concerning the weights of bodies will be true in like circumstances of other forces.
Weights are supposed to act in lines of direction parallel to each other. In fact, these lines are directed to the centre of the earth, but the angle formed between any two of them within the space occupied by a mechanical engine is so small, that the largest and most accurate astronomical instruments are scarcely capable of exhibiting it.
The simplest of those instruments, by means of which weights or forces are made to act in opposition to each other, are usually termedmechanical powers. Their names are, thelever, theaxisoraxle, andwheel, thepullyortackle, theinclined plane, thewedge, and thescrew.
Of the Lever.
The lever is defined to be a moveable and inflexible line, acted upon by three forces, the middle one of which is contrary in direction to the other two.
One of these forces is usually produced by the re-action of a fixed body, called thefulcrum.
If two contrary forces be applied to a lever at unequal distances from the fulcrum, they will equiponderate when the forces are to each other in the reciprocal proportion of their distances. For, by the resolution of force it appears, that if two contrary forces be applied to a straight lever, at distances from the fulcrum in the reciprocal proportion of their quantities, and in directions always parallel to each other, the lever will remain at rest in any position.
Since of the three forces which act on the lever, the two which are applied at the extremes, are always in a contrary direction to that which is applied in the space between them: this last force will sustain the effects of the other two; or, in other words, if the fulcrum be placedbetween the weights, it will be acted upon by their difference.
On the principle of the lever are made, scales for weighing different quantities of various kinds of things; the steelyard, which answers the same purpose by a single weight, removed to different distances from the fulcrum on a graduated arm, according as the body to be weighed is more or less in quantity; and the bent lever balance, which, by the revolution of a fixed weight, increasing in power as it ascends in the arc of a circle, indicates the weight of the counterpoise.
On this principle also, depend the motions of animals; the overcoming or lifting great weights by means of iron levers, called crows; the action of nutcrackers, pincers, and many other instruments of the same nature.
Of the Axis or Axle, and Wheel, and of the Pulley or Tackle.
The axis and wheel may be considered as a lever, one of the forces being applied at the circumference of the axis, and the other at the circumference of the wheel, the central line of the axis being as it were the fulcrum.
For if the semidiameter of the axis, be to the semidiameter of the wheel, reciprocally as the power of A is to the power B, the first of which is applied in the direction of a tangent of the axis, and the other in the direction of the tangent of the wheel, they will be in equilibrium.
To this power may be referred the capstan or crane, by which weights are raised; the winch and barrel, for drawing water, and numberless other machines on the same principle.
Thepullyis likewise explained on the same principle of the lever. Suppose the line A. C. to be a lever, whose arms A. B. and B. C. are equidistant from the fulcrum B. consequently the two equal powers E. and F. applied in the directions of the tangents to the circle in which the extremities are moveable, will be in equilibrium, and the fulcrum B. will sustain both forces.
But, suppose the fulcrum is at C. then a given force at E. will sustain in equilibrium a double force at F. for in that proportion reciprocally are their distances from the fulcrum. Whence it appears, that considering E. as a force, and F. as a weight to be raised, no increase of power is gained, when the pulley is fixed, but that a double increase of power is gained, when the pulley moves with the weight.
A combination of pullies is called a tackle, and a box containing one or more pullies, is called a block.
This is a tackle composed of four pullies, two of which are in the fixed block A. and the other two in the block B. that moves with the weight F. Now, because the rope is equally stretched throughout, each lower pulley will be acted upon by an equal part of the weight; and because in each pully that moves with the weight a double increase of power is gained; the force by which F. may be sustained will be equal to half the weight divided by the number of lower pullies: that is, as twice the number of lower pullies is toone, so is the weight suspending force.
But if the extremity of the rope C. be affixed to the lower block, it will sustain half as much as a pulley; consequently the analogy will then be, as twice the number of lower pullies, more 2 is to 1, so is the weight suspended to the suspending force.
The pulley or tackle is of such general utility, that it would seem unnecessary to point out any particular instance.
Of the inclined Plane, and of the Wedge.
The inclined plane has in its effects a near analogy to the lever; and the forces by which the same weight tends downwards in the directions of various planes, will be as the sines of their inclinations.
The wedge is composed of two inclined planes joined together at their common bases, in the direction of which the power is impressed.
This instrument is generally used in splitting wood, and was formerly applied in engines for stamping watch plates. The force impressed is commonly a blow, which is found to be much more effectual than a weight or pressure. This may be accounted for on the principles which obtain when resisting bodies are penetrated, as if the mass and velocity vary, the depths to which the impinging body penetrates will be in the compound ratio of the masses and the squares of the velocities.
All cutting instruments may be referred to the wedge. A chizel, or an axe, is a simple wedge; a saw is a number of chizels fixed in a line: a knife may be considered as a simple wedge, when employed in splitting; but if attention be paid to the edge, it is found to be a fine saw, as is evident from the much greater effect all knives produce by a drawing stroke, than what would have followed from a direct action of the edge.
Of the Screw, and of mechanical Engines, in general.
Thescrewis composed of two parts, one of which is called the screw, and consists of a spiral protuberance, called the thread, which is wound round a cylinder; and the other called the nut, is perforated to the dimensions of the cylinder, and in the internal cavity is cut a spiral groove adapted to receive the thread.
It would be difficult to enumerate the very many uses to which the screw is applied. It is extremely serviceable in compressing bodies together, as paper, linen, &c. It is the principal organ in all stamping instruments for striking coins, or making impressions on paper, linen, or cards, and is of vast utility to the philosopher, by affording an easy method of measuring or subdividing small spaces.A very ordinary screw will divide an inch into 5,000 parts; but the fine hardened steel screws, that are applied to astronomical instruments, will go much farther.
It is easy to conceive, that when forces applied to mechanical instruments are in equilibrium, if the least addition be made to one of them, it will preponderate and overcome the effort. But the want of a perfect polish or smoothness in the parts of all instruments, and the rigidity of all ropes, which increases with the tension, are great impediments to motion, and in compounded engines are found to diminish about one fourth of the effect of the power.
The properties of all the mechanical powers depending on the laws of motion, and the action or tendency to produce motion of each of the two forces, being applied in directions contrary to each other, the following general rule for finding the proportion of the forces in equilibrium on any machine will require no proof.
If two weights applied to the extremes of any mechanical engine, be to each other in the reciprocal proportion of the velocities resolved into a perpendicular direction, (rejecting the other part) which would be acquired by each when put in motion for the same indefinitely small time, they will be in equilibrio.
Whence it may be observed, that in all contrivances by which power is gained, a proportional loss is suffered in respect of time. If one man by means of a tackle, can raise as much weight, as ten men could by their unassisted strength, he will be ten times as long about it.
It is convenience alone, and not any actual increase of force, which we obtain from mechanics. As may be illustrated by the following example:
Suppose a man at the top of a house draws up ten weights, one at a time, by a single rope, in ten minutes: let him then have a tackle of five lower pullies, and he will draw up the whole ten at once with the same ease as he before raised up one; but in ten times the time, that is, in ten minutes. Thus we see the same work is performed in the same time, whether the tackle be used or not: but the convenience is, that if the whole ten weights be joined into one, they may be raised with the tackle, though it would be impossible to move them by the unassisted strength of one man; or suppose, instead of ten weights, a man draws ten buckets of water from the hold of a ship in ten minutes, and that the ship being leaky, admits an equal quantity in the same time. It is proposed that by means of a tackle, he shall raise a bucket ten times as capacious. With this assistance he performs it, but in as long a time as he required to draw the ten, and therefore is as far from gaining on the water in this latter case as in the former.
Since then no real gain of force is acquired from mechanical contrivances, there is the greatest reason to conclude, that a perpetual motion is not to be obtained. For in all instruments the friction of their parts, and other resistances, destroy a part of the moving force, and at last put an end to the motion.
Mechanical, inmathematics, denotes a construction of some problem, by the assistance of instruments, as the duplicature of the cube, and quadrature of the circle, in contradistinction to that which is done in an accurate and geometrical manner.
MECHE,Fr.SeeMatch.
MEDECIN,Fr.Physician.
MEDIATOR. Any state or power which interferes to adjust a quarrel between any two or more powers, is called a mediator.
MEDICINE-CHEST, is composed of all sorts of medicines necessary for a campaign, together with such chirurgical instruments as are useful, fitted up in chests, and portable. The army and navy are supplied with these at the expence of government.
Specific regulations have been issued by the war and navy offices, respecting the quantity and quality of the different medicines.
MEDIUM GUARD, a preparatory guard of the broad sword or sabre, which consists in presenting the sword in a perpendicular line with the centre of the opposed object, having the point upwards, the ward iron, and the cutting edge next the object.
MEER BUKSHY,Ind.Chief paymaster.
MEER TOZUK,Ind.A marshal whose business is to preserve order in a procession or line of march, and to report absentees.
MEGGHETERIARQUE,Fr.The commanding officer of a body of men, who formerly did duty at Constantinople, and were calledHéteriennes, being composed of soldiers that were enlisted in the allied nations.
MELEE,Fr.a military term, which is used among the French to express the hurry and confusion of a battle; thus,Un Général habile conserve sa tranquillité au milieu du combat, et dans l’horreur de la mêlée:—An able general preserves his presence of mind in the thickest of the battle, and remains calm during the whole of the conflict.Mêléecorresponds with the English expressionthick of the fight.
MEMOIRS, inmilitary literature, a species of history, written by persons who had some share in the transactions they relate, answering, in some measure, to what the Romans callcommentarii, i. e. commentaries. Hence Cæsar’s Commentaries, or the Memoirs of his Campaigns.
Memoiris the title given by military officers to those plans which they offer to their government or commanders on subjects relating to war or military economy.
MEMORIAL, an address to the government on any matter of public service.
BATTALION-MEN. All the soldiers belonging to the different companies of an infantry regiment are so called, except those of the two flank companies.
Camp-ColorMEN. Soldiers under the immediate command and direction of the quarter-master of a regiment. Their business is to assist in marking out the lines of an encampment, &c. to carry the camp colors to the field on days of exercise, and fix them occasionally for the purpose of enabling the troops to take up correct points in marching, &c. So that in this respect they frequently, indeed almost always, act as guides, or what the French callJalonneurs. They are likewise employed in the trenches, and in all fatigue duties.
Drag-ropeMEN. In the old artillery exercise, the men attached to light or heavy pieces of ordnance, for the purpose of advancing or retreating in action, were so called; the drag rope being exploded for the bricole, the term is preserved merely for explanation. The Frenchservans à la prolongeare of this description.
MENACE, an hostile threat. Any officer or soldier using menacing words or gestures in presence of a court-martial, or to a superior officer, is punishable for the same.—See theArticles of War.
MENSURATION, ingeneral, denotes the act or art of measuring lines, superficies, and solids.
Mensuration, inmilitary mathematics, is the art or science which treats of the measure of extension, or the magnitude of figures; and it is, next to arithmetic, a subject of the greatest use and importance, both in affairs that are absolutely necessary in human life, and in every branch of mathematics: a subject by which sciences are established, and commerce is conducted; by whose aid we manage our business, and inform ourselves of the wonderful operations in nature; by which we measure the heavens and the earth, estimate the capacities of all vessels and bulks of all bodies, gauge our liquors, build edifices, measure our lands and the works of artificers, buy and sell an infinite variety of things necessary in life, and are supplied with the means of making the calculations which are necessary for the construction of almost all machines.
It is evident that the close connection of this subject with the affairs of men would very early evince its importance to them; and accordingly the greatest among them have paid the utmost attention to it; and the chief and most essential discoveries in geometry in all ages, have been made in consequence of their efforts in this subject. Socrates thought that the prime use of geometry was to measure the ground, and indeed this business gave name to the subject; and most of the ancients seem to have had no other end besidesmensurationin view in all their labored geometrical disquisitions. Euclid’s elements are almost entirely devoted to it; and although there be contained in them many properties of geometrical figures, which may be applied to other purposes, and indeed of which the moderns have made the most material uses in various disquisitions of exceedingly different kinds; notwithstanding this, Euclid himself seems to have adapted them entirely to this purpose: for, if it be considered that his elements contain a continued chain of reasoning, and of truths, of which the former are successively applied to the discovery of the latter, one proposition depending on another, and the succeeding propositions still approximating towards some particular object near the end of each book; and when at the last we find that object to be the quality, proportion or relation between the magnitudes of figures both plane and solid; it is scarcely possible to avoid allowing this to have been Euclid’s grand object. And accordingly he determined the chief properties in the mensuration of rectilineal plane and solid figures; and squared all such planes, and cubed all such solids. The only curve figures which he attempted besides, are the circle and sphere; and when he could not accurately determine their measures, he gave an excellent method of approximating to them, by shewing how in a circle to inscribe a regular polygon which should not touch another circle, concentric with the former, although their circumferences should be ever so near together; and, in like manner, between any two concentric spheres to describe a polyhedron which should not any where touch the inner one: and approximations to their measures are all that have hitherto been given. But although he could not square the circle, nor cube the sphere, he determined the proportion of one circle to another, and of one sphere to another, as well as the proportions of all rectilineal similar figures to one another.
Archimedes took upmensurationwhere Euclid left it, and carried it a great length. He was the first who squared a curvilineal space, unless Hypocrates must be excepted on account of his lunes. In his times the conic sections were admitted in geometry, and he applied himself closely to the measuring of them as well as other figures. Accordingly he determined the relations of spheres, spheroids, and conoids, to cylinders and cones; and the relations of parabolas to rectilineal planes whose quadratures had long before been determined by Euclid. He hath left us also his attempts upon the circle: he proved that a circle is equal to a right angled triangle, whose base is equal to the circumference, and its altitude equal to the radius; and consequently that its area is found by drawing the radius into half the circumference; and so reduced the quadrature of the circle to the determination of the ratio of the diameter to the circumference; but which however hath not yet been done. Being disappointed ofthe exact quadrature of the circle, for want of the rectification of its circumference, which all his methods would not effect, he proceeded to assign an useful approximation to it: this he effected by the numerical calculation of the perimeters of the inscribed and circumscribed polygons; from which calculations it appears, that the perimeter of the circumscribed regular polygon of 192 sides is to the diameter in a less ratio than that of 3¹⁄₇ (3¹⁰⁄₇₀) to 1, and that the inscribed polygon of 96 sides is to the diameter in a greater ratio than that of 3¹⁰⁄₇₁ to 1; and consequently much more than the circumference of the circle is to the diameter in a less ratio than that of 3¹⁄₇ to 1, but greater than that of 3¹⁰⁄₇₁ to 1: the first ratio of 3¹⁄₇ to 1, reduced to whole numbers, gives that of 22 to 7, for 3¹⁄₇ : 1 ∷ 22 : 7, which therefore will be nearly the ratio of the circumference to the diameter. From this ratio of the circumference to the diameter he computed the approximate area of the circle, and found it to be to the square of the diameter as 11 to 14. He likewise determined the relation between the circle and elipsis, with that of their similar parts. The hyperbola too in all probability he attempted; but it is not to be supposed, that he met with any success, since approximations to its area are all that can be given by all the methods that have since been invented.
Besides these figures, he hath left us a treatise on the spiral described by a point moving uniformly along a right line, which at the same time moves with an uniform angular motion; and determined the proportion of its area to that of its circumscribed circle, as also the proportion of their sectors.
Throughout the whole works of this great man, which are chiefly onmensuration, he every where discovers the deepest design and finest invention; and seems to have been (with Euclid) exceedingly careful of admitting into his demonstrations nothing but principles perfectly geometrical and unexceptionable: and although his most general method of demonstrating the relations of curved figures to straight ones, be by inscribing polygons in them, yet to determine those relations, he does not increase the number and diminish the magnitude of the sides of the polygonad infinitum; but from this plain fundamental principle, allowed in Euclid’s elements, viz. that any quantity may be so often multiplied, or added to itself, as that the result shall exceed any proposed finite quantity of the same kind, he proves that to deny his figures to have the proposed relations, would involve an absurdity.
He demonstrated also many properties, particularly in the parabola, by means of certain numerical progressions, whose terms are similar to the inscribed figures: but without considering such series to be continuedad infinitum, and then summing up the terms of such infinite series.
He had another very curious and singular contrivance for determining the measures of figures, in which he proceeds, as it were, mechanically by weighing them.
Several other eminent men among the ancients wrote upon this subject, both before and after Euclid and Archimedes; but their attempts were usually upon particular parts of it, and according to methods not essentially different from theirs. Among these are to be reckoned Thales, Anaxagoras, Pythagoras, Bryson, Antiphon, Hypocrates of Chios, Plato, Apollonius, Philo, and Ptolomy; most of whom wrote of the quadrature of the circle, and those after Archimedes, by his method, usually extended the approximation to a greater degree of accuracy.
Many of the moderns have also prosecuted the same problem of the quadrature of the circle, after the same methods, to greater lengths: such are Viera, and Metius, whose proportion between the diameter and circumference is that of 113 to 355, which is within about ³⁄₁₀₀₀₀₀₀₀ of the true ratio; but above all, Ludolph van Ceulen, who with an amazing degree of industry and patience, by the same methods extended the ratio to 20 places of figures, making it that of 1 to3.14159265358979323846+.
The first material deviation from the principles used by the ancients in geometrical demonstrations was made by Cavalerius: the sides of their inscribed and circumscribed figures they always supposed of a finite and assignable number and length; he introduced the doctrine of indivisibles, a method which was very general and extensive, and which with great ease and expedition served to measure and compare geometrical figures. Very little new matter however was added to geometry by this method, its facility being its chief advantage. But there was great danger in using it, and it soon led the way to infinitely small elements, and infinitesimals of endless orders; methods which were very useful in solving difficult problems, and in investigating or demonstrating theories that are general and extensive; but sometimes led their incautious followers into errors and mistakes, which occasioned disputes and animosities among them. There were now, however, many excellent things performed in this subject; not only many new things were effected concerning the old figures, but new curves were measured; and for many things which could not be exactly squared or cubed, general and infinite approximating series were assigned, of which the laws of their continuation were manifest, and of some of which the terms were independent on each other. Mr. Wallis, Mr. Huygens, and Mr. James Gregory, performed wonders. Huygens in particular must be admired for his solid, accurate, and very masterly works.