The vse of the two mathematicall instruments the crosse staffe, (differing from that in common vse with the mariners:) and the Iacobs staffe: set foorth dialogue wise in two briefe and plaine treatises: the one most commodious for the mariner, and all such as are to deale in astronomicall matters: the other, profitable for the surueyor, to take the length, height, depth, or breadth, of any thing measurable. Set forth by Th. Hood. mathematicall lecturer in the citie of London. The staues are to be sold in Marke lane, at the house of Francis Cooke.

A Dialogue touching the vse of the Iacobs Staffe.

Scholler.

SInce my last departure from you, I haue diligently applied the crosse Staffe, accor∣ding vnto the rules which you haue pre∣scribed, and finde the same to be very con∣uenient for the purpose, which you haue set downe: that is, for the taking of the height of the Sunne and Starres, with their di∣stances: Yet can I not stay my selfe here, but must once againe presume vpon your courtesie.

Maister.

What is it that you desire? I loue not many words but you shall find me most ready to pleasure you, or a∣ny man els, as far forth as my simple skill will afford me.

Sc.

This is the matter: I would gladly learne the vse of the Iacobs Staffe as well as I haue learned the vse of the crosse staffe: May I not therefore so deuide the crosse staffe on the one side, that it may serue my turne for a Iacobs staffe, so that I be not troubled with the cariage of any more then one?

Ma.

Yes, very well. And loe, here is one so diuided, that as the one side of the Yard and Transame serueth for the Sun and Starres, so the other serueth for any dimension.

Sc.

This falleth out euen as I would haue it: seeing there∣fore the Staffe is ready at hand, if your leysure serue, I pray you instruct me in the vse thereof.

Ma.

With a good will: Moue you the questions, and I will answere them.

Sc.

Why doe they call it Iacobs Staffe? Was he the first inuentor of the thing.

Ma.

I know not that: but they take occasion to call it so, by reason of those wordes which are written Gen. 32.10. where the Patriarch sayth, That with his Staffe he came ouer Iordane: Wherein I thinke, they misconstrue his meaning. Notwith∣standing, by whom soeuer it was inuented, the Instrument questionlesse is of singular vse.

Sc.

Let that passe, and declare I pray you, what affinity this Staffe hath with the other. For comparisons will make a thing more plaine.

Ma.

First, as in the other Staffe, the degrées of the Yard and Transame are all one in number, so are the partitions in this. Secondly, as the yard and Transame in that, did ioyne together squarewise and at right angles, so must they doe in this. Thirdly, there must the same consideration be had in pla∣cing of the Vane in this, that was had in the placing of the Vane in the other: that is, it must stand euen with the begin∣ning of the partitions, except some speciall occasion require o∣therwise, as is to be séene in the measuring of breadthes. Fourthly, One of the two, either the Yard or the Transame, as occasion serueth, must be parallele vnto the thing measured, the other must be perpendicular.

Sc.

Is that a rule to be noted in the Iacobs staffe?

Ma.

Yea that it is: otherwise it is most certaine you com∣mit an errour.

Sc.

Then haue I béene amisse informed, as I haue séene men, that haue thought them selues to be of no small cunning in the vse of the Staffe, who measuring the height of a thing, haue made no matter how they held their Staffe, so that they did sée the toppe and the bottome of the thing to be measured, euen with the two ends of their Transame.

Ma.

Belaeue me in this, that it cannot be that their practise should be true. This one argument shall serue for a thousand. You know that the proofe of the vse of the Iacobs staffe stan∣deth vppon the proportion of Triangles. Now those Trian∣gles onely are proportionable which are like, and in like ma∣ner

situated, which they can neuer be except either the yard, or the Transame be parallel or perpendicular (as occasion ser∣ueth) to the thing measured.

Sc.

Your reason is good that it shoulde be so: but me thinketh that the perfourmaunce thereof should bee impos∣sible.

Ma.

Why so?

Sc.

You know that many Fields be full of Dales and hol∣low places: others be full of Hilles and Mountaines. More∣ouer, many things stand not vpright vpon the ground, but leane either forward or backeward, or else to the one side or to the other. Last of all in the bredth of a thing there is a great variety. For be it in a wall, or fielde, or what else whatsoe∣uer, there is for the most part either hollownesse or round∣nesse, or some such like thing which may make it impossible in my iudgement to draw either parallele or perpendicular line vnto them.

Ma.

I graunt those things to bee as you say: yet doe they not hinder a man from taking the measure of their length, height, or breadth. For where the place it selfe doeth not affoorde a leuelnesse: it is an easie matter to i∣magine it leuell. Thus also you must thinke with your selfe, that although Geometry bée the Arte of measuring well, and generally nothing may be excluded from the com∣passe thereof, yet must we not bee too curious in particu∣lars, least we shew our selues either ouer wise, or else too foolish. The Phisitian generally professeth to heale all disea∣ses, but when he commeth to deale with particular bodies, he faileth in his art, and yet is not to be discommended so long as he doeth that which Art requireth. It is enough for the Geometrician, if he giue the iust length of a streight line drawen betweene two places assigned. Or if hee can not doe it by one streight line, he may doe it by many. Whosoeuer immediately (for mediately he may finde out the length of a crooked line by meanes of a straight line)

whosoeuer, I say, requireth any more of him then that, he a∣buseth both the man and the art. Therefore, when any thing is propounded to be measured, be it length, height, or breadth, you are first of all to consider, whether the thing lie leuell, or stand vpright, or be straight, as it is required that it should be. If it be straight, you may presently fall in hand to measure it, as your rule shall direct you: If it be not straight, but be either crooked, or leaning one way or another, it must be your principall care, first to fit the same to your purpose: as for ex∣ample, in measuring the distance of a thing (for I will first be∣gin with that dimension, and will afterwards teach you what you shall doe in the rest) if the distaunce betwéene two places assigned be not straight, as you sée in these figures which fol∣low it is not, then choose out two points in the distance, the one being at the one end thereof, and the other in the other, be∣twéene which you may imagine a right line to be drawen, and then so hold your Staffe, that the yard or the Transame may be parallele, or perpendicular vnto it, as occasion shall serue.

Sc.

You haue throughly satisfied me in this. Tell me now wherein these staues do differ, séeing you haue declared wher∣in they doe agrée.

Ma.

Their difference consisteth in two things especially: in their partitions, and in their vse.

Sc.

Touching the first difference, it is euident indéed, that the parts of the other Staffe are vnequall, and that this is di∣uided into equall portions: but maketh it any matter howe many the parts are into which this Staffe is deuided?

Ma.

No: The more the better, and more precise will your worke be. And therfore, if you could diuide them into 100000. it were the more commendable, but 1000. sufficeth in this Staffe, which though it be not deuided into more, yet you may imagine it to be an hundred times as much as it is.

Sc.

What is the difference in the vse of the Staues?

Ma.

The generall difference is this: the other serueth for Astronomicall matters: this serueth for measuring of land.

Sc.

What meane you by those words, Measuring of land? Is it to take the breadth, or length only of a field? or may it be applied otherwise?

Ma.

The truth is this, that by this Staffe we may mea∣sure the length, the height, and the breadth of any thing, all the which we expresse by those words, Measuring of land: ha∣uing relation to the word Geometry, which signifieth the same if it be construed word for word.

Sc.

Before I doe come vnto the particular measuring of these thrée dimensions, length, height, and breadth, what gene∣rall notes haue I to obserue?

Ma.

First, that whether the Yard or the Transame be ap∣plied vnto your sight, it must stand close vnto the ball of your cheeke.

Sc.

But which end of the Yard or Transame must stand to the sight ward?

Ma.

That which hath the beginning of the degrées: for so it is most conuenient for the readinesse of your account. The second note is this: to winke with one eie. For the sight by that meanes will aime at your marke the better, when it is not parted but brought into one. The third obseruation is this: that, the distance of your marke be not ouer great.

Sc.

Why do you giue this note, considering that in the vse of the other Staffe, there was no limitation made of any di∣stance? For you measured therewith the starres in heauen.

Ma.

The case is not all one, because the Starres haue a brightnesse which helpeth their sight, and their beame com∣meth directly from the Centre of their body, vnto our eye. But in terrestriall things it is not so: For in them, by reason of the weakenesse of our sight, we may be greatly deceiued, ex∣cept they be within a reasonable compasse.

Sc.

What do you call a reasonable compasse, or distance?

Ma.

It cannot well be defined, because there are so many things required therein, that they will hardly iumpe all toge∣ther generally: as these, the clearenesse of the aire: the good∣nesse

of the sight: the largenesse of the instrument. The aire may be cleare, but the sight of a man may be dull. Againe, the sight may be quicke enough, but the aire may be thicke. A∣gaine, both the aire and sight may serue the turne, but the in∣strument which he is to vse may be so small, that he can not take any thing farre distant from him. So that we cannot ge∣nerally determine of the distance otherwise then thus, saying that that is a reasonable distaunce, which is proportionable both to our instrument and sight.

Sc.

Procéede I pray you in the matter wherein you were.

Ma.

The fourth regard must be had vnto your hands that they be steddy. For otherwise you may misse of the proporti∣on of your measure. The last note is this, to take heede vnto the place wherein you make your station.

Sc.

How shall I be sure of that?

Ma.

Thus: You know this, that either you stand vpright, or else leane one way or other. To be short, howsoeuer your situation be, whether it be standing, sitting, leaning, or vp∣right, that is the place of your station, which is right beneath your eie, so directly as is possible to be iudged and taken.

Sc.

These fiue things you say are generally to be noted. 1. The setting of the Saffe at the ball of the eye: 2. The wincking with one eie: 3. A reasonable distance: 4. A steddy hand: 5. The certaine place of your standing. Let vs now come vnto the particular vse of the Staffe in the thrée dimensions. You say, the Iacobs Staffe serueth to take the length, the height, and the breadth of any thing: Are these dimensions found out any more waies then one?

Ma.

Yea: The length, and the height, are found out thrée seuerall waies, and the breadth two waies. The first and the second way require one station: the last way in each dimen∣sion is performed by two stations.

Sc.

What meane you by a station? and what is it to per∣forme a thing by one or moe stations?

Ma.

A station is nothing else but the place wherein you

stand: to performe a thing therefore at one station, is nothing else but by a standing in one onely place by the helpe of your instrument, to take the measure of a thing. To performe it at two stations, is by standing in two seuerall places to take your measure.

Sc.

What generall note is there to be obserued, when we measure a thing at one station?

Ma.

You know this, that the measuring of a thing by the Iacobs Staffe, dependeth wholly vpon proportions, whose drift and finall end is this, by thrée numbers giuen, to find out the fourth. Two of these numbers are alwaies the partes of your Staffe, the one being the partes of your Transame, the other of the yard: the which numbers must be placed accor∣ding as the rules which follow shall direct you. The thirde number of proportion is some assigned measure in this man∣ner. If you measure the length of a thing, some certaine height must be assigned. If you measure the height or breadth of a thing, some certaine length, or distance must be giuen you for your thirde proportionall number, and this number must kéepe the third place in the rule of proportion, or Golden rule, or rule of thrée, as we do commonly terme it. Then must we say thus: that, as the partes of the Transame cutte off by the Yard, are vnto the parts of the Yard cut off by the Transame: Or contrariwise: As the parts of the yard cut off by the Tran∣same, are vnto the parts of the Transame cut off by the yard, so is the assigned measure which is giuen you, vnto the mea∣sure of the thing sought for. This is the chiefest note in mea∣suring at one station.

Sc.

What note is there to be obserued in measuring at two stations?

Ma.

There are thrée things specially to be noted: The first is, the making of your stations, that they be both of them in a right line with the thing to the which you leuell your staffe, whether you go forward or backward.

Sc.

How may that be done?

Ma.

Choose out some marke in the thing to be measured, which may be leuell with your eye, then set vp two staues where you minde to make your two stations, so that standing at the further Staffe, you may bring the other iust with the marke which you doe choose out: so shall you be sure that the two stations are in a right line with the marke chosen. For it is a generall rule, that if three pointes be in one leuell of the eye, those three pointes are in one right line. The second is, that your eye be no higher at the one station then it is at the o∣ther. This is to be done by the rule which I gaue you euen now For if you, standing at your second station, and holding your Staffe leuell from your sight, can sée the top of the Staffe which standeth at your first station, and also the marke which you choose, it is a certaine signe that your eye is in his iust height. The third note is, that the distuance betweene the two stations be diligently taken, and that in a right line, so neare as is possible: because the whole art of measuring by the Iacobs staffe, dependeth vpon a right line. This distance must be the third number of your proportion. And these be the chiefest ob∣seruations that I remember. If there be any thing else, I will put you in minde thereof as occasion serueth.

Sc.

Let vs come now vnto the particular dimensions, and beginne first with the length? What call you the lenth of the thing measured?

Ma.

It is the distaunce betwéene the place whereon you stand, and the place assigned, vnto which you direct your sight: and in this case we commonly say thus: How farre is it from my foote, to such or such a thing? as a Trée or House. &c.

Sc.

Giue me now the rules whereby I may direct my selfe in measuring any distance giuen.

Ma.

The first rule requireth, that the yard should be per∣pendicular to the length measured: And that is this. The yard of your Staffe being perpendicular vnto the thing measured. If your sight passe from the top of the yard ouer the end of the Transame to the farther end of the thing propounded, as the

partes of the yard cut off by the Transame, shall be vnto the parts of the Transame cut off by the yard, so shall the height of him that measureth bee vnto the length of the thing pro∣pounded.

Sc.

Here are two questions to be propounded. First, what call you the end of the Transame?

Ma.

That is generally called the end of the Transame, o∣uer the which your eie sight passeth wheresoeuer it be, whether it be in the very end of the wood it selfe, or any other place, where you list to set the Vane, so that your eye sight passe by that place.

Sc.

Secondly, what call you the height of him that mea∣sureth? For some man may be, and is a great deale higher then another.

Ma.

The height of the measure is the distaunce betwéene the top of the Staffe and the thing measured: as in the 1. figure from y. to o. and from a. to o. is the height of the measurer: so that whether he stand vppon the ground at the one end of the distance assigned, or whether he stand vpon some higher place, the maner of worke is all one: this being noted as a generall rule: That the height of the top of the yard from the ground, must be certainely knowen.

Sc.

Expresse the meaning of your Rule by some example.

Ma.

Put case the partes of the Yard cut off by the Tran∣same bee 60. and the partes of the Transame cut off by the Yard be 180. let the height of the Staffe from the ground be 4. foote: then I say that as 60. which are the partes of the Yard, are vnto 180. which are the parts of the Transame: e∣uen so is 4. féete, which is the supposed height of him that mea∣sureth vnto the length propounded. So that I conclude the length thereof to be 12. foote.

Sc.

This is easily concluded, and dependeth, as I remem∣ber vpon the 4. p. of the 6. booke of Euclide: where it is sayed that: In equiangle triangles, the sides which containe the equall Angles are proportionall, and the sides which are sub∣tended

vnder the equall Angles, are of like proportion. Moreo∣uer I perceiue (as you told me before) that the third number of the proportion, is the assigned height.

Ma.

It is true: and you may most manifestly perceiue that in the Figure which you sayd concerning Euclides propositi∣on: For the 2. Triangles a. e. i. and a. o. u. are like, because they are situated after one and the selfe same maner: therfore in the Triangle a. e. i. as a. e. is vnto a. i. so in the Triangle a. o. u. a. o. is vnto o. u. Thus you sée the commodity of the Staffe in measuring the length of a thing. And here you must note this, That it is no matter whether the thing measured do lie in a plaine or flat, or whether it be in the ascention or des∣cention of an Hill, or whether there be any concauity or con∣uexity betweene the ends of the thing measured. So that by this rule, we may measure the breadth of any Riuer, or of a∣ny Trench, or of any Valley betwéene the tops of the hils, or the distaunce of two Ships at sea &c.

Sc.

The first Rule required, that the yard should be perpen∣dicular vnto the thing measured. Put case the yard do lie pa∣rallele vnto it, as it is in the 2. Figure, what then?

Ma.

The yard being parallele vnto the thing measured, if your sight passe from the beginning of the yard ouer the ende of the Transame, then, as the parts of the Transame cut off by the yard, are vnto the parts of the yard cut off by the Transam, so is the height assigned vnto the length propounded to bee measured.

Sc.

So then by your Rule I conclude, that if the partes of the Transame cut off by the yard be 120. parts: and the parts of the yard cut off by the Transame be 210. and the height as∣signed and knowen be 400. féete, the length to be measured must be 700. féete, as I perceiue by the Golden rule, and by the proposition of Euclide mentioned before.

Ma.

You say very true: for considering the two Trian∣gles a. u. o. and a. i. e. are like, it must néedes follow, that as in the lesser Triangle o. u. is vnto u. a: so in the greater e. i.

must be vnto i. a.

Sc.

You sayd heretofore, that there were 3. waies to finde out the length of any thing, whereof 2. are already set downe: What is the third way?

Ma.

The two waies already set downe, require but one station. For you are not bound to remoue from the place whereon you first set your foot when you beginne to measure. But this third way requireth a double station, as in the 3. Fi∣gure, which is many times necessary, because that a Trée, or a Wall, or an Hill, may be betwéene our sight and the end of the thing, whose distaunce from vs we desire to know, so that our sight cannot come vnto the end thereof. The third way dif∣fereth from the former in this, that there is no altitude to be assigned and giuen for the third number of the proportion, but the third number of the proportion, is the distaunce betwéene the two stations.

Sc.

I pray you answere me to this: Considering that I must take 2. stations, at which is it best to begin? Is it best to begin my first station at the end of the distance propounded, and so go backeward? or is it best to begin my first station a good way from the end of the distaunce propounded, and then come for∣ward to make my second station?

Ma.

That is neither here nor there: But the common vse is to begin the first station, at the end of the distance propoun∣ded, and so go backward: in the which thing doing, you must take héede of that especially, which I told you before, that you goe backeward in a straight line: so that the line imagined to be drawen from the one end of the distance propounded vnto the other, being continued vnto the second station must make but one right line: otherwise you shall erre in your measure more or lesse.

Sc.

What distaunce is best to be made betwéene the two stations?

Ma.

The larger the better: For it is good neuer to take a short station, if you may haue a large one: because your er∣rour

will be the lesse thereby.

Sc,

Giue me now the rule which I must follow.

Ma.

The Yard being parallele vnto the length propoun∣ded to be measured, and your eye at each seuerall station, pas∣sing from the head thereof ouer the end of the Transame, vn∣to something that standeth vpright, at the farther end of the distaunce prounded: as the difference of the greater segment of your Transame shall be vnto the lesser segment: so shall the length betweene the two stations be vnto the distance pro∣pounded.

Sc.

In this kinde of measuring, must the Transame kéepe one place vpon the yard, without pulling it nigher, or putting it farther off from the sight?

Ma.

The rule the I gaue you euen now requireth that it should do so. And then you must note this, that the farther you go backeward from your first station, the lower must the Transame sinke into his socket.

Sc.

Giue me an example of that which you haue sayd.

Ma.

Put case that at your first station the partes of the Transame cut off by the Yard were 108. at your second stati∣on, put case the partes of the Transame cut off by the Yard were 72. imagine also the distance betwéene your two stations were 40. féete: I must séeke out the difference betwéene the se∣uerall partes of the Transame cut off by the Yard: which by subduction I finde to be 36. (for that is the remainder, 72. be∣ing subducted out of 108.) whereupon I conclude thus, that as the difference of the greater segment of the Transame, name∣ly, 36. is vnto the lesser segment, which is 72. so is the distance betwéene my two stations, to the length propounded: and ther∣fore the length propunded to be measured is 80. féete. For if 36. giue 72. then 40. féete (which is the distance of the two sta∣tions) must néeds yeeld 80. Thus haue you hitherto séene the thrée seuerall waies how to measure the length of any thing propounded: It followeth now to instruct you in taking the height of any thing-propounded.

Sc.

What call you the height of a thing?

Ma.

The height of a thing, is the perpendicular drawen from the top thereof, vnto the base whereon it standeth.

Sc.

Then you regard not at all the crookednesse of a thing.

Ma.

No: I care not at all for the fashion thereof. For whe∣ther it leane forward or backeward, to the right hand or to the left, I care not so that from the top thereof I may draw down a perpendicular line: as you sée in the fourth figure.

Sc.

Do you take all this line?

Ma.

Your measure taketh no more then is from the leuell of your eie vpward: and therfore this must be a generall rule, that, if the thing propounded to be measured, doe stand leuell with your feete, you must adde vnto the height found out, the height also of your owne stature.

Sc.

How many waies is the height of a thing measured?

Ma.

Thrée maner of waies, euen as the length or distance was.

Sc.

What must bee the third proportionall number in the first two waies?

Ma.

As in the measuring of lengths, the height assigned and giuen, was the thirde proportionall number; so in the measuring of heights, the length assigned is the third number of the proportion:

Sc.

Giue me the rules to direct me by.

Ma.

The first rule is this: Your Transame making a right angle with the height to be measured: and your sight passing from the beginning thereof to the toppe of the height: as the partes of the Transame cut off by the Yard, are to the parts of the Yard cut off by the Transame, so is the length giuen vn∣to the height.

Sc.

Put case then the parts of the Transame were 60. and the partes of the Yard 36. and the length giuen 20. féete: the height according to the rule must be 12. féete.

Ma.

It is so: But you must remember to adde the height of your owne stature: so that it shall be in all 16, féete, or 15. féete

according as you shall please to allow for your owne height.

Sc.

Let me request this by the way: is there no rule for the measuring of depths?

Ma.

It is all one with the measuring of heights. For the depth is nothing else but a rehearsed height. And therefore in the measuring of the depth of a thing, The Yard of your staffe being parallele vnto the side of the thing, whose depth you measure, and your sight passing from the toppe thereof: as the parts of the Transame cut off by the Yard, are to the parts of the Yard cut off by the Transame, so is the assigned length vn∣to the depth of the thing.

Sc.

I perceiue your meaning. As for example: If the parts of the Transame be 5. and the partes of the Yard 13. and the diameter of the well (which is here in this example the assg∣ned length) be 10. féete long, the depth must be 26. féete.

Ma.

It is true, if you account from the top of your Staffe, and therefore that you may haue the iust depth of the thing, you must take away from the aforesaid number as much as is aboue the thing measured: which if you suppose to be one foote in length, then is the well 25. foote déepe, as appeareth in the fifth figure.

Sc.

Thus much by the way concerning the measuring of any depth. What other rule haue you for the measuring of heights?

Ma.

The first rule required that the Transame should be perpendicular vnto the height propounded. The second requi∣reth the Yard to be perpendicular: as in the sixt figure: so that it differeth from the other onely in the manner of holding the Staffe. The rule is this: The yard of your Staffe being per∣pendicular vnto the thing measured as the parts of the yard are to the parts of the Transame, so is the assigned length vnto the height propounded to be measured.

Sc.

So that I may conclude thus. If the parts of the Yard be 60. and the parts of the Transame 60. and the assigned length be 250. féete: the height propounded must also be 250. féete.

Ma.

Your conclusion is allowable: But I must put you in minde of this, to adde the allowance of your stature, which if you suppose to be 4. féete, then is the whole height 254. féete: Now marke this one thing for your further learning.

Sc.

What is that?

Ma.

By this meane you may finde out the seuerall partes of any height propunded: as in the seuenth figure you may finde out how many féete are contained betweene the top and the nether side of the window: or generally betweene any parts else, or placing as many Vanes vpon the Transame, as you would measure seuerall parts.

Sc.

I must so place my Vanes, that from the end of the yard I may see each part propounded.

Ma.

You say right, the rule then must be this: The Yard being perpendicular, as it was before vnto the height pro∣pounded, as the parts of the yard cut off by the Transame are vnto the distance of the Vanes, so is the whole height, vnto the part thereof propounded.

Sc.

Let me sheld that I vnderstand you, by propounding an example: I say, that if the partes of the Transame be 50. and the diffaunce of the Vanes be 20. partes: and the whole height of the thing measured be 37. feete, then the part pro∣pounded must be 14. feete, and 〈◊〉〈◊〉.

Ma.

You vnderstand it well. And here is yet another thing to be noted: that if you know the length of any one peece of the thing whose height you desire to know, it is sufficient to bring you to the knowledge of the whole height, as in the 8. figure.

Sc.

What rule haue you for that to direct me in the worke?

Ma.

The Vanes of your staffe must be placed as they were before, ••••••ely, in such sort that your sight passing from the end of the yard by the Vanes, you may see both the ends of the part whose length you know. The rule then is this: that as the distance of the two Vanes is vnto the remnant of the Transam downe vnto the yard, so is the part known vnto the remnant of the height propounded. So that if the distance of the Vanes

be 20. parts, and the re••inant of the Transame from the ne∣thermost Vane downe ward be 30. parts: and the part of the height knowen be 14. féete, and ⅘ I conclude the remnant of the height to be 22. féete, and ⅕.

Sc.

That standeth with reason, and may be proued by the example next going before this. For if the whole height were 37. féete, and the part propounded 14. féete, 〈…〉〈…〉 that part being taken out of the whole, declareth the remnant to be 22 féete, and ⅕.

Ma.

The former two rules measured the height of a thing at one station The third rule requireth a double ••••ation, as in the ninth figure. The rule is this: The Yard of your Staffe being perpendicular to the thing measured; as the difference of the two segments of the yard is to the distance betweene the two stations, so are the partes of the Transame cut off by the Yard to the height propounded.

Sc.

What meane you by the difference of the two segments of the Yard?

Ma.

You may easily know what that meaneth, by that which hath beene taught you before, in measuring the diffance or length of a thing at two stations. Yet I will make it plaine vnto you here: When you make your first station, you 〈◊〉〈◊〉 the Transame must cut the Yarde in one place or other: also when you make your second station, the ••••••same must cut the Yard in another place, not the same which was before: that distance or parts of the Yard, contained bett••••me those 〈◊〉〈◊〉 se∣uerall places, are called the differente of the segnm••••s.

Sc.

In this case I must 〈…〉〈…〉 and downe.

Ma.

No but you must mo••e it only either 〈…〉〈…〉 your sight.

Sc.

When must it come toward my 〈…〉〈…〉 it goe farther from it?

Ma.

That fulleth not according 〈…〉〈…〉 If in 〈…〉〈…〉 from the

thing measured, your Transame must goe farther from your fight: If you come nigher, your Transame also must come nigher your fight. And here I giue you warning againe, that you goe forward and backward in a right line, directly vppon the thing propounded.

Sc.

Let me here also by example shew that I vnderstand your Rule. You say that as the difference betwéene the two segments of the yard is vnto the distance betwéene the two stations, so are the partes of the Transame vnto the heighth. Whereupon I conclude thus: that if the difference of the two segments be 23. and the distance of the two stations be 30. ••••••se, and the parts of the Transame cut off by the yard be 44. then the heighth must be 57. féete, and 9/23. as in the Figure.

Ma.

Hitherto you haue learned how to measure the length and the height of things: now it followeth to measure the breadth.

Sc.

Giue me leaure, I pray you, in this place to moue thrée questions: and first what meane you by the breadth?

Ma.

The breadth here to be measured, is no more, then the length of the right line, conteyned betweene the two ends of the thing propounded, without any consideration of the round∣nesse or hollownesse of the thing.

Sc.

Secondly, how must I stand to hold my Staffe?

Ma.

You must stand so, that your staffe being applied vnto your eie, it may be perpendicular vnto the bredth propounded.

Sc.

Last of all, when I purpose to measure the breadth of a∣ny thing, as of a wall, or such like, against which part of it must I stand? against the iust middest of it? or against the one end? or else where I please, betwéene the two endes? or without both the ends?

Ma.

It is at your owne choise.

Sc.

In measuring the former dimensions, my sight did passe but by one end of the Transame, must it do so likewise in this?

Ma.

That is according as you stand to take the measure.

If you stand within the two ends of the thing propounded to be measured, your sight must passe by the two ends of the Trā∣same. If you stand against the one end of the thing, or cleane without it, your sight passeth but by one end of the Transame.

Sc.

What meane you here by the ends of the Transame? are they not here also to be taken as they were before?

Ma.

Yes: The one end of the Transame must alwaies he that, where the partitions begin: the other end is that, vppon which the Vane must stand, and by which the sight passeth. For many times it will so fall out that your sight cannot passe by both the ends of the wood, so that you shall be enforced to place the Vane in some one part or other of the Transame, that it may be a limit for your sight. And this is a generall rule, that if you stand within the ends of the breadth propounded, the one end of your Transame must be on the one side of the yard, and the other on the other side.

Sc.

But will the yard be iust betwéene them both?

Ma.

No: Except your station be made accordingly. If you stand in the middest of the thing to be measured, then will the yard be iust in the middest, betwéene the endes of the Transame: if you stand more to the one end then to the o∣ther, the yard also will fall out in like manner, more toward the one end of the Transame, then toward the other.

Sc.

How many waies are there to he vsed to measure the breadth by?

Ma.

We may measure it two waies: either by the helpe of one station, or of two.

Sc.

If I measure it by the helpe of one station, what must be knowen for the third proportionall number?

Ma.

The distance betweene you and the thing propounded.

Sc.

Put case that it be hollow inward betwéene the two ends thereof: or swell outwardly: to which part must I then measure?

Ma.

If it be conuexe, or round outward, the best way to measure the breadth is by two stations: but if it be hollow in∣ward,

you must measure no further then from your féet to the right line, which runneth betwéene the two ends of the thing propounded.

Sc.

What rule is there for the measuring of a breadth at one station?

Ma.

This. The Yard being perpendicular vnto the breadth assigned, as the parts of the Yard cut off by the Transame are vnto the partes of the Transame contained betweene the two ends, so is the distance betweene your feete and the thing measured, to the breadth thereof. As in the tenth figure.

Sc.

I perceiue it well: For there are here two proportionall triangles cut in péeces with a line in the midst: Therefore as in the triangle A B C. the line B C. is the line A C. so is the line B F. in the greater triangle to the line F G. which is the breadth of the wall. This dependeth vpon that which went be∣fore. Now what is your rule for the measuring at two statiōs.

Ma.

It is this: The Yard being perpendicular vnto the thing measured, & your sight passing by the ends of the Tran∣same (as it is required:) as the difference of the two segments of the yard is vnto the distaunce betweene the two stations, so are the parts of the Transame betweene the two ends thereof, vn∣to the breadth propounded: as it is in the 11. figure.

Sc.

This also is very easie by reason of that which was taught before. So that I may conclude, that if the difference of the two segments in the yard be 30. parts: and the distance of my two stations be 40. féete: and the parts betwéene the two ends of the Transame be 50. then must the bredth of the thing be 66. féete and ⅔. Haue you any other thing to be added?

Ma.

No: But that I would haue you to practise this dili∣gently, and I will hereafter further you in other things.

Ma.

I thanke you for it. Yet before I depart, I pray you let me aske you one question: You said before, that in measuring the breadth of any thing it is at my choise to stand where I will, either within the two ends of the thing to be measured, or against the one end thereof, or without both the endes, in

what place I shall thinke most expedient. This last kinde of measuring me thinketh is strange (as for the other I perceiue well enough how their proportion will fall out) wherefore I pray you helpe me herein.

Ma.

Set the question as you shall thinke it best for your owne vnderstanding.

Sc.

Put case I were to take the breadth of a riuer, as in the 12. figure, vnto whose brinke I can not come any nigher then 40. féete, by reason of some marish grounds adioyning there∣vnto: what meanes shall I vse to finde the breadth thereof?

Sc.

Put case I were to take the breadth of a riuer, as in the 12. figure, vnto whose brinke I can not come any nigher then 40. féete, by reason of some marish grounds adioyning there∣vnto: what meanes shall I vse to finde the breadth thereof?

Sc.

But how must they stand in respect of the riuer?

Ma.

Indéed that is a speciall thing to be noted: They must stand so, that the right line imagined to come from the two markes, may make a right angle with the riuers side: other∣wise you shall erre in taking of your measure: Because that only is accounted the breadth of a thing, which maketh a right angle with the length.

Sc.

Hauing chosen out these two markes, what is then to be done?

Ma.

You must set vp another, which must stand also iust in one right line with the two first markes, whose distaunce from the mark on this side the riuer, must either be giuen you, as being already certainely knowne: or else it must be found out by the first rule concerning the measuring of breadths.

Sc.

Put case I either know it, or finde it to be 40. féete.

Ma.

You must choose out a conuenient place, wherein you may make your station, & settle your selfe about your worke.

Sc.

How must that place lie from the marke that was last set vp?

Ma.

It must lie from it so, that the line drawne from that

marke vnto the place of your standing, may make a right an∣gle with the line drawne betweene the thrée afornamed markes.

Sc.

Must I not know the iust length betwéene the place wherein I make my station, and the marke which I set vp.

Ma.

Yes: For it is the third proportionall number.

Sc.

Imagine that distance to be 60. féete.

Ma.

Then your rule to finde out the breadth is this: The Yard being perpendicular with the marke set vp, and your sight passing from the head thereof by the end of the Transame vnto the marke standing on the farther side of the riuer, as the parts of the Yard cut off by the Transame, are vnto the partes of the Transame cut off by the Yarde: so is the distaunce be∣tweene your station and the marke set vp vnto the distaunce betweene the sayd marke, and the marke standing vppon the further side of the riuer.

Sc.

But what is this to the question that I propounded? you make an aunswere concerning the distaunce of the two markes, whereas I moued a question of the breadth of the Riuer.

Ma.

Be not angry, I will satisfie you straight waies: first finde out the distance of these two markes.

Sc.

That can I easily doe: For the parts of the yard being 30. and the parts of the Transame 36. and the distance of my station from the first marke 60. féete. The first marke must be distant from the third 72. féete.

Ma.

Then what is the breadth of the riuer?

Sc.

That must come from you, for I cannot make the con∣clusion.

Ma.

Then marke this. You sayd the distaunce of the first marke from the third being on the farther side of the Riuer is 72. féete: and the distaunce of the first marke from the second being on the hither side of the Riuer, was graunted to be 40. féete: then take 40. out of 72. and the remainder being 32. féete, declareth the breadth of the Riuer.

Sc.

I perceiue now what a commodity it is to haue an in∣structor. For the thing which séemed vnto me most strange, and altogether impertinent to my question, is by your meanes made most evident vnto me, and greatly to the matter pro∣pounded. But must I take this course, whensoeuer I am dri∣uen to stand without the compasse of the thing measured.

Ma.

Yea, that you must: And sometimes also you shall be driuen to the second rule, wherein two stations are requi∣red, which soeuer you vse it commeth all vnto one purpose, and will eastly affoord you the breadth of a thing.

Sc.

I can but thanke you for your paines, and good will, in accepting me for your scholler.

Ma.

And I likewise you, for making choise of me to be your maister. My will is good to helpe whom I may: howsoe∣uer others may make more boast of their cunning.

FINIS.