Читать книгу The Way To Geometry - Petrus Ramus - Страница 3

1. Geometry is the Art of measuring well.

The end or scope of Geometry is to measure well: Therefore it is defined of the end, as generally all other Arts are. To measure well therefore is to consider the nature and affections of every thing that is to be measured: To compare such like things one with another: And to understand their reason and proportion and similitude. For all that is to measure well, whether it bee that by Congruency and application of some assigned measure: Or by Multiplication of the termes or bounds: Or by Division of the product made by multiplication: Or by any other way whatsoever the affection of the thing to be measured be considered.

But this end of Geometry will appeare much more beautifull and glorious in the use and geometricall workes and practise then by precepts, when thou shalt observe Astronomers, Geographers, Land-meaters, Sea-men, Enginers, Architects, Carpenters, Painters, and Carvers, in the description and measuring of the Starres, Countries, Lands, Engins, Seas, Buildings, Pictures, and Statues or Images to use the helpe of no other art but of Geometry. Wherefore here the name of this art commeth farre short of the thing meant by it. (For Geometria, made of Gè, which in the Greeke language signifieth the Earth; and Métron, a measure, importeth no more, but as one would say Land-measuring. And Geometra, is but Agrimensor, A land-meter: or as Tully calleth him Decempedator, a Pole-man: or as Plautus, Finitor, a Marke-man.) when as this Art teacheth not only how to measure the Land or the Earth, but the Water, and the Aire, yea and the whole World too, and in it all Bodies, Surfaces, Lines, and whatsoever else is to bee measured.

Now a Measure, as Aristotle doth determine it, in every thing to be measured, is some small thing conceived and set out by the measurer; and of the Geometers it is called Mensura famosa, a knowne measure. Which kinde of measures, were at first, as Vitruvius and Herodo teache us, taken from mans body: whereupon Protagoras sayd, That man was the measure of all things, which speech of his, Saint Iohn, Apoc. 21. 17. doth seeme to approve. True it is, that beside those, there are some other sorts of measures, especially greater ones, taken from other things, yet all of them generally made and defined by those. And because the stature and bignesse of men is greater in some places, then it is ordinarily in others, therefore the measures taken from them are greater in some countries, then they are in others. Behold here a catalogue, and description of such as are commonly either used amongst us, or some times mentioned in our stories and other bookes translated into our English tongue.

Granum hordei, a Barley corne, like as a wheat corne in weights, is no kinde of measure, but is quiddam minimum in mensura, some least thing in a measure, whereof it is, as it were, made, and whereby it is rectified.

Digitus, a Finger breadth, conteineth 2. barly cornes length, or foure layd side to side:

Pollex, a Thumbe breadth; called otherwise Vncia, an ynch, 3. barley cornes in length:

Palmus, or Palmus minor, an Handbreadth, 4. fingers, or 3. ynches.

Spithama, or Palmus major, a Span, 3. hands breadth, or 9. ynches.

Cubitus, a Cubit, halfe a yard, from the elbow to the top of the middle finger, 6. hands breadth, or two spannes.

Ulna, from the top of the shoulder or arme-hole, to the top of the middle finger. It is two folde; A yard and an Elne. A yard, containeth 2. cubites, or 3. foote: An Elne, one yard and a quarter, or 2. cubites and ½.

Pes, a Foot, 4. hands breadth, or twelve ynches.

Gradus, or Passus minor, a Steppe, two foote and an halfe.

Passus, or Passus major, a Stride, two steppes, or five foote.

Pertica, a Pertch, Pole, Rod or Lugge, 5. yardes and an halfe.

Stadium, a Furlong; after the Romans, 125. pases: the English, 40. rod.

Milliare, or Milliarium, that is mille passus, 1000. passes, or 8. furlongs.

Leuca, a League, 2. miles: used by the French, spaniards, and seamen.

Parasanga, about 4. miles: a Persian, & common Dutch mile; 30. furlongs.

Schœnos, 40. furlongs: an Egyptian, or swedland mile.

Now for a confirmation of that which hath beene saide, heare the words of the Statute.

It is ordained, That 3. graines of Barley, dry and round, do make an Ynch: 12. ynches do make a Foote: 3. foote do make a Yard: 5. yardes and ½ doe make a Perch: And 40. perches in length, and 4. in breadth, doe make an Aker: 33. Edwar. 1. De terris mensurandis: & De compositione ulnarum & Perticarum.

Item, Bee it enacted by the authority aforesaid; That a Mile shall be taken and reckoned in this manner, and no otherwise; That is to say, a Mile to containe 8. furlongs: And every Furlong to containe 40. lugges or poles: And every Lugge or Pole to containe 16. foote and ½. 25. Eliza. An Act for restraint of new building, &c.

These, as I said, are according to diverse countries, where they are used, much different one from another: which difference, in my judgment; ariseth especially out of the difference of the Foote, by which generally they are all made, whether they be greater of lesser. For the Hand being as before hath beene taught, the fourth part of the foot whether greater or lesser: And the Ynch, the third part of the hand, whether greater or lesser.

Item, the Yard, containing 3. foote, whether greater or lesser: And the Rodde 5. yardes and ½, whether greater or lesser, and so forth of the rest; It must needes follow, that the Foote beeing in some places greater then it is in other some, these measures, the Hand, I meane, the Ynch, the Yard, the Rod, must needes be greater or lesser in some places then they are in other. Of this diversity therefore, and difference of the foot, in forreine countries, as farre as mine intelligence will informe me, because the place doth invite me, I will here adde these few lines following. For of the rest, because they are of more speciall use, I will God willing, as just occasion shall be administred, speake more plentifully hereafter.

Of this argument divers men have written somewhat, more or lesse: But none to my knowledge, more copiously and curiously, then Iames Capell, a Frenchman, and the learned Willebrand, Snellius, of Leiden in Holland, for they have compared, and that very diligently, many and sundry kinds of these measures one with another. The first as you may see in his treatise De mensuris intervallorum describeth these eleven following: of which the greatest is Pes Babylonius, the Babylonian foote; the least, Pes Toletanus, the foote used about Toledo in Spaine: And the meane betweene both, Pes Atticus, that used about Athens in Greece. For they are one unto another as 20. 15. and 12. are one unto another. Therefore if the Spanish foote, being the least, be devided into 12. ynches, and every inch againe into 10. partes, and so the whole foote into 120. the Atticke foote shall containe of those parts 150. and the Babylonian, 200. To this Atticke foote, of all other, doth ours come the neerest: For our English foote comprehendeth almost 152. such parts.

The other, to witt the learned Snellius, in his Eratosthenes Batavus, a booke which hee hath written of the true quantity of the compasse of the Earth, describeth many more, and that after a farre more exact and curious manner.

Here observe, that besides those by us here set downe, there are certaine others by him mentioned, which as hee writeth are found wholly to agree with some one or other of these. For Rheinlandicus, that of Rheinland or Leiden, which hee maketh his base, is all one with Romanus, the Italian or Roman foote. Lovaniensis, that of Lovane, with that of Antwerpe: Bremensis, that of Breme in Germany, with that of Hafnia, in Denmarke. Onely his Pes Arabicus, the Arabian foote, or that mentioned in Abulfada, and Nubiensis: the Geographers I have overpassed, because hee dareth not, for certeine, affirme what it was.

Looke of what parts Pes Tolitanus, the spanish foote, or that of Toledo in Spaine, conteineth 120. of such is the Pes.

Heidelbergicus, that of Heidelberg, 137.

Hetruscus, that of Tuscan, in Italie, 138.

Sedanensis, of Sedan in France, 139.

Romanus, that of Rome in Italy, 144.

Atticus, of Athens in Greece, 150.

Anglicus, of England, 152.

Parisinus, of Paris in France, 160.

Syriacus, of Syria, 166.

Ægyptiacus, of Egypt, 171.

Hebraicus, that of Iudæa, 180.

Babylonius, that of Babylon, 200.

Looke of what parts Pes Romanus, the foote of Rome, (which is all one with the foote of Rheinland) is 1000. of such parts is the foote of

Toledo, in Spaine, 864.

Mechlin, in Brabant, 890.

Strausburgh, in Germany, 891.

Amsterdam, in Holland, 904.

Antwerpe, in Brabant, 909.

Bavaria, in Germany, 924.

Coppen-haun, in Denmarke, 934.

Goes, in Zeland, 954.

Middleburge, in Zeland, 960.

London, in England, 968.

Noremberge, in Germany, 974.

Ziriczee, in Zeland, 980.

The ancient Greeke, 1042.

Dort, in Holland, 1050.

Paris, in France, 1055.

Briel, in Holland, 1060.

Venice, in Italy, 1101.

Babylon, in Chaldæa, 1172.

Alexandria, in Egypt, 1200.

Antioch, in Syria, 1360.

Of all other therefore our English foote commeth neerest unto that used by the Greekes: And the learned Master Ro. Hues, was not much amisse, who in his booke or Treatise De Globis, thus writeth of it Pedem nostrum Angli cum Græcorum pedi æqualem invenimus, comparatione facta cum Græcorum pede, quem Agricola & alij ex antiquis monumentis tradiderunt.

Now by any one of these knowne and compared with ours, to all English men well knowne the rest may easily be proportioned out.

2. The thing proposed to bee measured is a Magnitude.

Magnitudo, a Magnitude or Bignesse is the subject about which Geometry is busied. For every Art hath a proper subject about which it doth employ al his rules and precepts: And by this especially they doe differ one from another. So the subject of Grammar was speech; of Logicke, reason; of Arithmeticke, numbers; and so now of Geometry it is a magnitude, all whose kindes, differences and affections, are hereafter to be declared.

3. A Magnitude is a continuall quantity.

A Magnitude is quantitas continua, a continued, or continuall quantity. A number is quantitas discreta, a disjoined quantity: As one, two, three, foure; doe consist of one, two, three, foure unities, which are disjoyned and severed parts: whereas the parts of a Line, Surface, and Body are contained and continued without any manner of disjunction, separation, or distinction at all, as by and by shall better and more plainely appeare. Therefore a Magnitude is here understood to be that whereby every thing to be measured is said to bee great: As a Line from hence is said to be long, a Surface broade, a Body solid: Wherefore Length, Breadth, and solidity are Magnitudes.

4. That is continuum, continuall, whose parts are contained or held together by some common bound.

This definition of it selfe is somewhat obscure, and to be understand onely in a geometricall sense: And it dependeth especially of the common bounde. For the parts (which here are so called) are nothing in the whole, but in a potentia or powre: Neither indeede may the whole magnitude bee conceived, but as it is compact of his parts, which notwithstanding wee may in all places assume or take as conteined and continued with a common bound, which Aristotle nameth a Common limit; but Euclide a Common section, as in a line, is a Point, in a surface, a Line: in a body, a Surface.

5. A bound is the outmost of a Magnitude.

Terminus, a Terme, or Bound is here understood to bee that which doth either bound, limite, or end actu, in deede; as in the beginning and end of a magnitude: Or potentia, in powre or ability, as when it is the common bound of the continuall magnitude. Neither is the Bound a parte of the bounded magnitude: For the thing bounding is one thing, and the thing bounded is another: For the Bound is one distance, dimension, or degree, inferiour to the thing bounded: A Point is the bound of a line, and it is lesse then a line by one degree, because it cannot bee divided, which a line may. A Line is the bound of a surface, and it is also lesse then a surface by one distance or dimension, because it is only length, wheras a surface hath both length and breadth. A Surface is the bound of a body, and it is lesse likewise then it is by one dimension, because it is onely length and breadth, whereas as a body hath both length, breadth, and thickenesse.

Now every Magnitude actu, in deede, is terminate, bounded and finite, yet the geometer doth desire some time to have an infinite line granted him, but no otherwise infinite or farther to bee drawane out then may serve his turne.

6. A Magnitude is both infinitely made, and continued, and cut or divided by those things wherewith it is bounded.

A line, a surface, and a body are made gemetrically by the motion of a point, line, and surface: Item, they are conteined, continued, and cut or divided by a point, line, and surface. But a Line is bounded by a point: a surface, by a line: And a Body by a surface, as afterward by their severall kindes shall be understood.

Now that all magnitudes are cut or divided by the same wherewith they are bounded, is conceived out of the definition of Continuum, e. 4. For if the common band to containe and couple together the parts of a Line, surface, & Body, be a Point, Line, and Surface, it must needes bee that a section or division shall be made by those common bandes: And that to bee dissolved which they did containe and knitt together.

7. A point is an undivisible signe in a magnitude.

A Point, as here it is defined, is not naturall and to bee perceived by sense; Because sense onely perceiveth that which is a body; And if there be any thing lesse then other to be perceived by sense, that is called a Point. Wherefore a Point is no Magnitude: But it is onely that which in a Magnitude is conceived and imagined to bee undivisible. And although it be voide of all bignesse or Magnitude, yet is it the beginning of all magnitudes, the beginning I meane potentiâ, in powre.

8. Magnitudes commensurable, are those which one and the same measure doth measure: Contrariwise, Magnitudes incommensurable are those, which the same measure cannot measure. 1, 2. d. X.

Magnitudes compared betweene themselves in respect of numbers have Symmetry or commensurability, and Reason or rationality: Of themselves, Congruity and Adscription. But the measure of a magnitude is onely by supposition, and at the discretion of the Geometer, to take as pleaseth him, whether an ynch, an hand breadth, foote, or any other thing whatsoever, for a measure. Therefore two magnitudes, the one a foote long, the other two foote long, are commensurable; because the magnitude of one foote doth measure them both, the first once, the second twice. But some magnitudes there are which have no common measure, as the Diagony of a quadrate and his side, 116. p. X. actu, in deede, are Asymmetra, incommensurable: And yet they are potentiâ, by power, symmetra, commensurable, to witt by their quadrates: For the quadrate of the diagony is double to the quadrate of the side.

9. Rationall Magnitudes are those whose reason may bee expressed by a number of the measure given. Contrariwise they are irrationalls. 5. d. X.

Ratio, Reason, Rate, or Rationality, what it is our Authour (and likewise Salignacus) have taught us in the first Chapter of the second booke of their Arithmetickes: Thither therefore I referre thee.

Data mensura, a Measure given or assigned, is of Euclide called Rhetè, that is spoken, (or which may be uttered) definite, certaine, to witt which may bee expressed by some number, which is no other then that, which as we said, was called mensura famosa, a knowne or famous measure.

Therefore Irrationall magnitudes, on the contrary, are understood to be such whose reason or rate may not bee expressed by a number or a measure assigned: As the side of the side of a quadrate of 20. foote unto a magnitude of two foote; of which kinde of magnitudes, thirteene sorts are mentioned in the tenth booke of Euclides Elements: such are the segments of a right line proportionally cutte, unto the whole line. The Diameter in a circle is rationall: But it is irrationall unto the side of an inscribed quinquangle: The Diagony of an Icosahedron and Dodecahedron is irrationall unto the side.

10. Congruall or agreeable magnitudes are those, whose parts beeing applyed or laid one upon another doe fill an equall place.

Symmetria, Symmetry or Commensurability and Rate were from numbers: The next affections of Magnitudes are altogether geometricall.

Congruentia, Congruency, Agreeablenesse is of two magnitudes, when the first parts of the one doe agree to the first parts of the other, the meane to the meane, the extreames or ends to the ends, and lastly the parts of the one, in all respects to the parts, of the other: so Lines are congruall or agreeable, when the bounding, points of the one, applyed to the bounding points of the other, and the whole lengths to the whole lengthes, doe occupie or fill the same place. So Surfaces doe agree, when the bounding lines, with the bounding lines: And the plots bounded, with the plots bounded doe occupie the same place. Now bodies if they do agree, they do seeme only to agree by their surfaces. And by this kind of congruency do we measure the bodies of all both liquid and dry things, to witt, by filling an equall place. Thus also doe the moniers judge the monies and coines to be equall, by the equall weight of the plates in filling up of an equall place. But here note, that there is nothing that is onely a line, or a surface onely, that is naturall and sensible to the touch, but whatsoever is naturall, and thus to be discerned is corporeall.

Therefore

11. Congruall or agreeable Magnitudes are equall. 8. ax. j.

A lesser right line may agree to a part of a greater, but to so much of it, it is equall, with how much it doth agree: Neither is that axiome reciprocall or to be converted: For neither in deede are Congruity and Equality reciprocall or convertible. For a Triangle may bee equall to a Parallelogramme, yet it cannot in all points agree to it: And so to a Circle there is sometimes sought an equall quadrate, although incongruall or not agreeing with it: Because those things which are of the like kinde doe onely agree.

12. Magnitudes are described betweene themselves, one with another, when the bounds of the one are bounded within the boundes of the other: That which is within, is called the inscript: and that which is without, the Circumscript.

Now followeth Adscription, whose kindes are Inscription and Circumscription; That is when one figure is written or made within another: This when it is written or made about another figure.

Homogenea, Homogenealls or figures of the same kinde onely betweene themselves rectitermina, or right bounded, are properly adscribed betweene themselves, and with a round. Notwithstanding, at the 15. booke of Euclides Elements Heterogenea, Heterogenealls or figures of divers kindes are also adscribed, to witt the five ordinate plaine bodies betweene themselves: And a right line is inscribed within a periphery and a triangle.

But the use of adscription of a rectilineall and circle, shall hereafter manifest singular and notable mysteries by the reason and meanes of adscripts; which adscription shall be the key whereby a way is opened unto that most excellent doctrine taught by the subtenses or inscripts of a circle as Ptolomey speakes, or Sines, as the latter writers call them.