The Way To Geometry

The ninth Booke, of P. Ramus Geometry, which intreateth of the measuring of right lines by like right-angled triangles

The Geometry of like right-angled triangles, amongst many other uses that it hath, it doth especially afford us the geodæsy or measuring of right lines: And that mastery, which before (at the 2 e viij) attributed the right angled triangles, shall here be found to be a true mastery indeed. For it shall containe the geodesy of right lines; and afterward the geodesy of plaines and solides, by the measuring of their sides, which are right lines.

1. For the measuring of right lines; we will use the Iacobs staffe, which is a squire of unequall shankes.

Radius, commonly called Baculus Iacob, Iacobs staffe, as if it had been long since invented and practised by that holy Patriarke, is a very auncient instrument, and of all other Geometricall instruments, commonly used, the best and fittest for this use. Archimedes in his book of the Number of the sand, seemeth to mention some such thing: And Hipparchus, with an instrument not much unlike this, boldly attempted an haynous matter in the sight of God, as Pliny thinketh, namely to deliver unto posterity the number of the starres, and to assigne or fixe them in their true places by the Norma, the squire or Iacobs staffe. And indeed true it is that the Radius is not onely used for the measuring of the earth and land: But especially for the defining or limiting of the starres in their places and order: And for the describing and setting out of all the regions and waies of the heavenly city. Yea and Virgill the famous Poet, in his 3 Ecloge, Ecquis fuit alter, Descripsit radio totum, qui gentibus orbem? and againe afterward in the 6 of his Eneiades, hath noted both these uses. Cœliquè meatus. Describent radio & surgentia sidera dicent. Long after this the Iewes and Arabians, as Rabbi Levi; But in these latter daies, the Germaines especially, as Regiomontanus; Werner, Schoner, and Appian have grac'd it: But above all other the learned Gemma Phrisius in a severall worke of that argument onely, hath illustrated and taught the use of it plainely and fully.

The Iacobs staffe therefore according to his owne, and those Geometricall parts, shall here be described (The astronomicall distribution wee reserve to his time and place.) And that done, the use of it shall be shewed in the measuring of lines.

This instrument, at the discretion of the measurer may be greater or lesser. For the quantity of the same can no otherwayes be determined.

2. The shankes of the staffe are the Index and the Transome.

The principall parts of this instrument are two, the Index, or Staffe, which is the greater or longer part: and the Transversarium, or Transome, and is the lesser and shorter.

3. The Index is the double and one tenth part of the transome.

Or thus: The Index is to the transversary double and 1/10 part thereof. H. As here thou seest.

4. The Transome is that which rideth upon the Index, and is to be slid higher or lower at pleasure.

Or, The transversary is to be moved upon the Index, sometimes higher, sometimes lower: H. This proportion in defining and making of the shankes of the instrument is perpetually to be observed: as if the transome be 10. parts, the Index must be 21. If that be 189. this shall be 90. or if it be 2000. this shall be 4200. Neither doth it skill what the numbers be, so this be their proportion. More than this, That the greater the numbers be, that is the lesser that the divisions be, the better will it be in the use. And because the Index must beare, and the transome is to be borne; let the index be thicker, and the transome the thinner.

But of what matter each part of the staffe be made, whether of brasse or wood it skilleth not, so it be firme, and will not cast or warpe. Notwithstanding, the transome will more conveniently be moved up and downe by brasen pipes, both by it selfe, and upon the Index higher or lower right angle wise, so touching one another, that the alterne mouth of the one may touch the side of the other. The thrid pipe is to be moved or slid up and downe, from one end of the transome to the other; and therefore it may be called the Cursor. The fourth and fifth pipes, fixed and immoveable, are set upon the ends of the transome, are unto the third and second of equall height with finnes, to restraine when neede is, the opticke line, and as it were, with certaine points to define it in the transome.

The three first pipes may, as occasion shall require, be fastened or staied with brasen scrues. With these pipes therefore the transome may be made as great, as need shall require, as here thou seest.

The fabricke or manner of making the instrument hath hitherto beene taught, the use thereof followeth: unto which in generall is required: First, a just distance. For the sight is not infinite. Secondly, that one eye be closed: For the optick faculty conveighed from both the eyes into one, doth aime more certainely; and the instrument is more fitly applied and set to the cheeke bone, then to any other place. For here the eye is as it were the center of the circle, into which the transome is inscribed. Thirdly, the hands must be steady; for if they shake, the proportion of the Geodesy must needes be troubled and uncertaine. Lastly, the place of the station is from the midst of the foote.

5. If the sight doe passe from the beginning of one shanke, it passeth by the end of the other: And the one shanke is perpendicular unto the magnitude to be measured, the other parallell.

These common and generall things are premised. That the sight is from the beginning of the Index by the end of the transome; Or contrariwise, From the beginning of the transome, unto the end of the Index. And that the Index is right, that is, perpendicular to the line to be measured, the transome parallell. Or contrariwise. Now the perpendicularity of the Index, in measurings of lengthts, may be tried by a plummet of lead appendent; But in heights and breadths, the eye must be trusted; although a little varying of the plummet can make no sensible errour. By the end of the transome, understand that which is made by the line visuall, whether it be the outmost finne, or the Cursour in any other place whatsoever.

6. Length and Altitude have a threefold measure; The first and second kinde of measure require but one distance, and that by granting a dimension of one of them, for the third proportionall: The third two distances, and such onely is the dimension of Latitude.

Geodesy of right lines is two fold; of one distance, or of two. Geodesy of one distance is when the measurer for the finding of the desired dimension doth not change his place of standing. Geodesy of two distances is when the measurer by reason of some impediment lying in the way betweene him and the magnitude to be measured, is constrained to change his place, and make a double standing.

Here observe, That length and heighth, may be joyntly measured both with one, and with a double station: But breadth may not be measured otherwise than with two.

7. If the sight be from the beginning of the Index right or plumbe unto the length, and unto the farther end of the same, as the segment of the Index is, unto the segment of the transome, so is the heighth of the measurer unto the length.

Let therefore the segment of the Index, from the toppe, I meane, unto the transome be 6. parts. The segment of the transome, to wit, from the Index unto the opticke line be 18. The Index, which here is the heighth of the measurer, 4. foote: The length, by the rule of three, shall be 12. foote. The figure is thus, for as ae, is to ei, so is ao, unto ou, by the 12. e vij. For they are like triangles. For aei, and aou, are right angles: And that which is at a is common to them both: Wherefore the remainder is equall to the remainder, by the 4. e vij.

The same manner of measuring shall be used from an higher place; as out of y, the segment of the Index is 5. parts; the segment of the transome 6: and then the height be 10 foote: the same Length shall be found to bee 12 foote.

Neither is it any matter at all, whether the length in a plaine or levell underneath: Or in an ascent or descent of a mountaine, as in the figure under written.

Thus mayest thou measure the breadths of Rivers, Valleys, and Ditches. For the Length is alwayes after this manner, so that one may measure the distance of shippes on the Sea, as also Thales Milesius, in Proclus at the 26 p j, did measure them. An example thou hast here.

Hereafter in the measuring of Longitude and Altitude, sight is unto the toppe of the heighth. Which here I doe now forewarne thee of, least afterward it should in vaine be reitered often.

The second manner of measuring a Length is thus:

8. If the sight be from the beginning of the index parallell to the length to be measured, as the segment of the transome is, unto the segment of the index, so shall the heighth given be to the length.

As if the segment of the Transome be 120 parts: the height given 400 foote: The segment of the Index 210 parts: The length, by the golden rule shall be 700 foote. The figure is thus. And the demonstration is like unto the former; or indeed more easier. For the triangles are equiangles, as afore. Therefore as ou is to ua: so is ei to ia.

This is the first and second kinde of measuring of a Longitude, by one single distance or station: The third which is by a double distance doth now follow. Here the transome, if there be roome enough for the measurer to goe farre enough backe, must be put lower, in the second distance.

9. If the sight be from the beginning of the transverie parallell to the length to be measured, as in the index the difference of the greater segment is unto the lesser; so is the difference of the second station unto the length.

This kinde of Geodæsy is somewhat more subtile than the former were. The figure is thus; in which let the first ayming be from a, the beginning of the transome, and out of ai the length sought by o, the end of the Index, unto e, the toppe of the heighth: And let the segment of the Index be ou: The second ayming let it be from y, the beginning of the transome, out of a greater distance by s, the end of the Index, unto e, the same note of the heighth: And let the segment of the Index be sr.

Here the measuring performed, is the taking of the difference betweene ou and sr. The rest are faigned onely for demonstrations sake. Therefore in the first station let aml, be from the beginning of the transome, be parallell to ye. Here first mu, is equall to sr. For the triangles mua, and sry, are equall in their shankes ua, and ry, by the grant (Because the transome standeth still in his owne place:) And the angles at mua, uam, are equall to the angles: And all right angles are equall, by the 14 e iij. These are the outter and inner opposite one to another: And such are equall by the 1 e v. Therefore they are equilaters, by the 2 e vij; And om, is the difference of the segments of the Index. Then as om is to mu, so is el, to li; as the equation of three degrees doth shew. For, by the 12 e vij, as om is to ma: so is el to la: And as ma is to mu; so is la, to li. Therefore by right, as om, is to mu: so is el, to li: And by the 12 e vj, so is ya, to ai: As if the difference of the first segment be 36 parts: The second segment be 72 parts: The difference of the second station 40 foote. The length sought shall be 80 foote. And here indeed is no heighth definitely given, that may make any bound of the principall proportion. Notwithstanding the Heighth, although it be of an unknowne measure, is the bound of the length sought: And therefore it is an helpe and meanes to argue the question. Because it is conceived to stand plumbe upon the outmost end of the length.

Therefore that third kinde of measuring of length is oftentimes necessary, when by neither of the former wayes the length may possibly be taken, by reason of some impediment in the way, to wit of a wall, or tree, or house, or mountaine, whereby the end of the length may not be seene, which was the first way: Nor an height next adjoyning to the end of the length is given, which is the second way.

Hitherto we have spoken of the threefold measure of longitude, the first and second out of an heighth given the third cut of a double distance: The measuring of heighth followeth next, and that is also threefold. Now heighth is a perpendicular line falling from the toppe of the magnitude, unto the ground or plaine whereon the measurer doth stand, after which manner Altitude or heighth was defined at the 9 e iiij. The first geodesy or manner of measuring of heighths is thus.

10. If the sight be from the beginning of the transome perpendicular unto the height to be measured, as the segment of the transome, is unto the segment of the Index, so shall the length given be to the height.

Let the segment of the transome be 60 parts: the segment of the Index 36: the Length given 120 foote: the height sought shall be, by the golden rule, 72 foote.

The Figure is thus: And the demonstration is by the 12 e vij, as afore: but here is to be added the height of the measurer; which if it be 4 foot, the whole height shall be 76 foote.

Therefore in an eversed altitude

11. If the sight be from the beginning of the Index parallell to the height, as the segment of the transome is, unto the segment of the index, so shall the length given be, unto the height sought.

Eversa altitudo, An eversed altitude (Reversed, H:) is that which we call depth, which indeed is nothing else, in the Geometers sense, but heighth turned topsie turvie, as we say, or with the heeles upward. For out of the heighth concluded by subducting that which is above ground, the heighth or depth of a Well shall remaine.

Let the segment of the transome ae, be 5 parts: the segment of the Index ei, be 13: the diameter of the Well (which now standeth for the length:) be 10 foote, which at toppe is supposed to be equall to that at bottome: the opposite height, by the 12 e vij, and the golden rule shall be 26 foote: From whence you must take the segment of the Index reaching over the mouth of the Well: And the true height (or depth) shall remaine; as if that segment of 13 parts be as much as 2 foote, the height sought shall be 24 foote. The second manner of measuring of heights followeth.

12. If the sight be from the beginning of the Index perpendicular to the heighth to be measured, as the segment of the Index is unto the segment of the Transome, so shall the length given be to the heighth.

As if the segment of the Index be 60 parts: and the segment also of the transome be 60: And the Length given be 250 foote: By the Rule of three, the height also shall be 250 foote: as thou seest in the example underneath: For as ae is to ei; so is aeo to ou, by the 12 e vij. But here unto the height found, you must adde the height of the measurer: Which if it be 4 foot, the whole height shall be 254 foote.

Therefore

13. If the sight be from the beginning of the Index (perpendicular to the magnitude to be measured) by the names of the transome, unto the ends of some known part of the height, as the distance of the Names is, unto the rest of the transome above them, so shall the known part be unto the part sought.

Or thus: If the sight passe from the beginning of the Index being right, by the vanes of the transversary, to the tearmes of some parts; as the distance of the vanes is unto the rest of the transversary above the index, so is the part knowne unto the remainder: H.

This is a consectary of a knowne part of an height, from whence the rest may be knowne, as in the figure.

As ou is unto uy, so is ei to is. For as ou, is unto ua: so is ei unto ia, by the 12 e vij. And as ua, is to uy, so is ia unto is; and by right, as ou, is to uy, so is ei, to is. Here thou hast three bounds of the proportion. Let therefore ou, be 20 parts: uy 30: And ei, the knowne part, let it be 15 foote: Therefore thou shalt conclude is, the rest to be 22½.

The first and second kinde of measuring of heights is thus: The third followeth.

14 If the sight be from the beginning of the Index perpendicular to the heighth, as in the Index the difference of the segment, is unto the difference of the distance or station; so is the segment of the transome unto the heighth.

Hitherto you must recall that subtilty, which was used in the third manner of measuring of lengths.

Let the first aime be taken from a, the beginning of the Index perpendicular unto the height to be measured: And from an unknowne length ai, by o, the end of the transome, unto e, the toppe of the height ei: And let the segment of the Index be ua. The second ayme, let it be taken from y, the beginning of the same Index; and out of a greater distance, by s, the end of the transome, unto the same toppe e. And the segment of the Index let it be ry.

Here, as afore, the measuring is performed and done, by the taking of the difference of the said yr, above au: Now the demonstration is concluded, as in the former was taught. Let the parallell lsm, be erected against aoe.

Here first the triangles oua, & srl, are equilaters, by the 2 e vij.; (seeing that the angles at a, and l, the externall and internall, are equall in bases ou, and sr, for the segment in each distance is the same still:) Therefore ua, is equall to rl. Now the rest is concluded by a sorites of foure degrees: As yr, is unto yi: so by the 12. e vij. is sr, that is, ou, unto ei: And as ou, is unto ei, so is au, that is, lr, unto ai. Therefore the remainder yl, unto the remainder ya; shall be as yr, is unto the whole yi, and therefore from the first unto the last, as sr, is to ei.

Therefore let the difference of the Index be 23. parts: The difference of the distance 30. foote: The segment of the transome 44. parts: The height shall be 57.9/23. or foote.

Therefore

15 Out of the Geodesy of heights, the difference of two heights is manifest.

Or thus: By the measure of one altitude, we may know the difference of two altitudes: H.

For when thou hast taken or found both of them, by some one of the former wayes, take the lesser out of the greater; and the remaine shall be the heighth desired. From hence therefore by one of the towers of unequall heighth, you may measure the heighth of the other. First out of the lesser, let the length be taken by the first way: Because the height of the lesser, wherein thou art, is easie to be taken, either by a plumbe-line, let fall from the toppe to the bottom, or by some one of the former waies. Then measure the heighth, which is above the lesser: And adde that to the lesser, and thou shalt have the whole heighth, by the first or second way. The figure is thus, and the demonstration is out of the 12. e vij. For as ae, is to ei, so is ao, to ou. Contrariwise out of an higher Tower, one may measure a lesser.

16 If the sight be first from the toppe, then againe from the base or middle place of the greater, by the vanes of the transome unto the toppe of the lesser heighth; as the said parts of the yards are unto the part of the first yard; so the heighth betweene the stations shall be unto his excesse above the heighth desired.

Let the unequall heights be these, as, the lesser, and uy, the greater: And out of the assigned greater uy, let the lesser, as, be sought. And let the sight be first from u, the toppe of the greater, unto a, the toppe of the lesser, making at the shankes of the staffe the triangle urm. Then againe let the same sight be from the base, or from the lower end of uy, the heighth given, unto a, the same toppe of the lesser, making by the shankes of the staffe the triangle yln, so that the segments of the yard be, the upper one, I meane, ur, the neather one ul: I say the whole of ur, and nl, is unto ur: so is the uy, greater heighth assigned, unto as, the lesser sought.

The Demonstration, by drawing of ao, a perpendicular unto uy, is a proportion out of two triangles of equall heighth. For the forth of the totall equally heighted triangles uao, and yas, although they be reciprocall in situation, they have their bases uo, and as, as if their were oy. Then they have the same with the whole triangles; as also the subducted triangles urm, and ynl, of equal heighth; to wit whose common heighth is the segment of the transome remained still in the same place, there rm, here yl. And therefore the bases of these, namely, the segments of the yards ur, and nl, have the same rate with uo, unto oy. As therefore uo, is unto oy: so is, ur, unto nl. And backward, as nl, is to ur; so is, yo, unto ou, as here thou seest:

Therefore furthermore by composition of the Antecedent with the Consequent unto the Consequent, by the 5 c 9 ij. Arith. As nl, and ur, are unto ur: so are yo, and ou, unto ou, that is yu, unto ou, on this manner.

there is given nl, and ur, for the first proportionall: ur, for the second: and yu, for the third: Therefore there is also given ou, for the fourth: Which ou, subducted out of uy, there remaineth oy, that is, as, the lesser altitude sought.

For let the parts of the yard be 12. and 6. and the summe of them 18. Now as 18. is unto 12. so is the whole altitude uy, 190. foote, unto the excesse 126⅔ foote. The remainder therefore 63⅓ foote, shall be as, the lesser heighth sought.

But thou maist more fitly dispose and order this proportion thus: As ur, is unto nl: so is uo unto oy. Therefore by Arithmeticall composition, as ur, and nl, are unto nl: so uo, and oy, that is, the whole uy, is unto oy, that is, unto as. For here a subduction of the proportion, after the composition is no way necessary, by the crosse rule of societia, thus:

The second station might have beene in o, the end of the perpendicular from a. But by taking the ayme out of the toppe of the lesser altitude, the demonstration shall be yet againe more easie and short, by the two triangles at the yard aei, and aef, resembling the two whole triangles aou, and aoy, in like situation, the parts of the shanke cut, are on each side the segments of the transome.

One may againe also out of the toppe of a Turret measure the distance of two turrets one from another: For it is the first manner of measuring of longitudes, neither doth it here differ any whit from it, more than the yard is hang'd without the heighth given. The figure is thus: And the Demonstration is by the 12. e vij. For as ae, the segment of the yard, is unto ei the segment of the transome: so is the assigned altitude ao, unto the length ou.

The geodesy or measuring of altitude is thus, where either the length, or some part of the length is given, as in the first and second way: Or where the distance is double, as in the third.

17 If the sight be from the beginning of the yard being right or perpendicular, by the vanes of the transome, unto the ends of the breadth; as in the yard the difference of the segment is unto the differēce of the distance, so is the distance of the vanes unto the breadth.

The measuring of breadth, that is of a thwart or crosse line, remaineth. The Figure and Demonstration is thus: The first ayming, let it be aei, by o, and u, the vanes of the transome ou. The second, let it be yei, by s, and r, the vanes of the transome sr. Then by the point s, let the parallell lsm, be drawne against aoe. Here first, the triangles oua, and sil, are equilaters, by the 2 e vij. Because the angles at n and j, are right angles: And uao, and jls, the outter and inner, are equall in their bases ou, and sj, by the grant: Because here the segment of the transome remaineth the same: Therefore ua, is equall to jl. These grounds thus laid, the demonstration of the third altitude here taken place. For as yl, is unto ya: so is sj, unto er: And, because parts are proportionall unto their multiplicants, so is sr, unto ei: for the rest doe agree.

The same shall be the geodesy or manner of measuring, if thou wouldest from some higher place, measure the breadth that is beneath thee, as in the last example. But from the distance of two places, that is, from latitude or breadth, as of Trees, Mountaines, Cities, Geographers and Chorographers do gaine great advantages and helpes.

Wherefore the geodesy or measuring of right lines is thus in length, heighth, and breadth, from whence the Painter, the Architect, and Cosmographer, may view and gather of many famous place the windowes, the statues or imagery, pyramides, signes, and lastly, the length and heighth, either by a single or double: the breadth by a double dimension onely, that is, they may thus behold and take of all places the nature and symmetry; as in the example next following thou mayst make triall when thou pleasest.