Thesaurarium mathematicae, or, The treasury of mathematicks containing variety of usefull practices in arithmetick, geometry, trigonometry, astronomy, geography, navigation and surveying ... to which is annexed a table of 10000 logarithms, log-sines, and log-tangents / by John Taylor.

CHAP. X. Of MEASURING, Of Board, Glass, Tiling, Paving, Timber, Stone, and Irregular Solids, such as Geo∣metry can give no Rule for the Measuring thereof.

SECT. I. Of the Measuring of Board, Glass, Paving, Tiling, &c.

I Have already in the fourth Chapter of this Book, and the second Section thereof, ap∣plyed Geometry to the finding out of the Su∣perficial Content of all Regular Superficies. I have also in the ninth Chapter, and the third Section thereof, shewed how the Superficial Content of any Irregular Superficies may be found, by redu∣cing

them into Regular Forms: which I have explained amply in that Section, I shall there∣fore here be as plain and brief as is possible.

PROP. I. To Measure a Piece of Board, Plank, Glass, &c.

In Measuring of Board, Glass, &c. Carpen∣ters and other Mechanicks measure by the Foot, 12 Inches unto the Foot; so that a Foot of Board, or Glass, contains 144 Square Inches.

Now if a Piece of Board, Plank, or Glass, be required to be measured, let it be either a Parallelogram, or Tapering Piece: first by the Rules aforegoing find the Content thereof in Inches, and that Product divide by 144, the Quotient is the Content of that Superficies in Feet.

PROP. II. To measure Tiling, Flooring, Roofing, and Parti∣tioning-works.

In Tiling, Flooring, Roofing, and Partitioning∣work, Carpenters, and other Workmen, reckon by the Square, which is 10 Feet every way; so that a Square containeth 100 Feet: Exam∣ple.

There is a Roof 14 Feet broad, what length thereof shall make a Square? Divide 100 by 14, it yields 7 1/7 Feet.

Now if you have any Number of Feet gi∣ven, and the Number of Squares therein contai∣ned are required, divide that Number by 100, the product is Squares.

PROP. III. To measure Paving, Plaistering, Wainscotting, and Painting-work.

In Paving, Plaistering, Wainscotting, and Painting-work, Mechanicks reckon by the Yard Square, so each Yard is equal unto 9 Square Feet.

By the Rules aforegoing find the Superficial Content of the Court, Alley, &c. in Feet: which divide by 9, the Quotient is the Number of Yards in that work contained.

SECT. II. Of the Measuring of Timber, Stone, and Irregular Solids.

IN Superficial Measure a Superficial Foot con∣tains 144 Square Inches; but in Solid Measure a Foot contains 1728 Cubick Inches. Now ha∣ving already in the fourth Chapter of this Book, and the third Section thereof, largely applyed Geometry unto the Measuring of all

Regular Solids, I shall therefore in this Place be as brief as possible, only I shall be somewhat larger in the Mensuration of Irregular Solids, which is of special Moment in sundry parts of the Mathematical Practices.

PROP. I. How to Measure any kind of Timber, or Stone, whether Three-square, Four-square, Many-square, Round, or of any other fashion, provided it be streight and equal all along.

To perform which first by the Rules afore∣going in Chap. 4. §. 2. get the Superficial Content at the End, and then say,

As 144, the Inches of the Superficial Content of the End of a Cubick Foot,

To a Cubick Foot containing 1000 parts;

So is the Superficial Content of the End of any piece of Timber,

To the Solid Content of one Foot length of the said piece of Timber.

According to which Mr. Phillips calculated the ensuing Table, which I have thought fit hereunto to annex.

Case 2 Or the solid Content in Feet, &c. may be found otherwise thus.

By the Rules aforegoing find the Content of the End of the piece of Timber in Inches, which Content multiply by the length of the said piece of Timber, or Stone in Inches, and that Product divide by 1728, it produceth the Solid

Content of that Piece of Timber, or Stone, in Feet, and parts of a Foot.

A Table shewing the Solid Content of one Foot-length of any Piece of Timber, according to the Superfi∣cial Content at the End thereof.

Feet.	Parts.	Feet.	Parts.
The Inches of the Content at the End.	1	0	007	The Inches of the Content at the End.	200	1	398
2	0	014	300	2	083
3	0	021	400	2	778
4	0	028	500	3	472
5	0	035	600	4	167
6	0	042	700	4	861
7	0	049	800	5	556
8	0	056	900	6	250
9	0	062	1000	6	944
10	0	069	2000	13	888
20	0	139	3000	20	833
30	0	208	4000	27	778
40	0	278	5000	34	722
50	0	347	6000	41	666
60	0	417	7000	48	711
70	0	485	8000	55	555
80	0	556	9000	62	500
90	0	625	10000	69	444
100	0	694	20000	138	888

PROP. II. To measure Round Timber which is Hollow: or a∣ny other Hollow Body.

If Hollow Timber be to be measured, first measure the Stick as though it were not Hollow, then find the Solidity of the Concavity, as though it were Massie Timber, then substract this last found Content, out of the whole Content be∣fore found, the remainder is the Content of that Hollow Body.

PROP. III. To Measure Tapering Timber, or Stone.

Those Tapering Bodies are either Segments of Cones, or Pyramids: now the way to measure such bodies, is demonstrated in Prop. the 4. and 5. §. 3. Chap. 4: But now to find the Content of these Segments do thus: measure the Solidity of the whole Cone, or Pyramid, and then find the Content of the Top part thereof cut off, (as if it were a Cone, or Pyramid of it self) and the Content thereof, deduct from the Content of the whole Cone, or Pyramid: so shall the re∣mainder be the Content of the Segment requi∣red: which reduced into Feet gives the Solid Content of that Piece of Timber in Feet. Now to find the length of the Top part cut off, from the Cone, or Pyramid, say,

As the Difference of the breadth of the two Ends, To the length between them:

So is the breadth of the greater End, To the whole length of the Cone, or Pyramid.

PROP. IV. How to find the Solid Content of any Solid Body, in any strange form, such as Geometry can given: no Rule for the measuring thereof.

These strange forms are either Branches in Metal, Crowns, Cups, Bowles, Pots, Screws, or Twisted Ballisters,* 1.1 or any other Irregular-Solid, that keep not in thick∣ness one Quantity, but are thicker in one place, than in another, so that no man by Geometry, is possible to measure their Solidity.

Now for the finding the Content of any such like Irregular Body in Inches or Feet, do thus: Cause to be made a Hollow Cube, or Parallelepipe∣don, so that you may measure it with an Inch∣Rule without Difficulty, and so to know the true Content of the whole, or any part thereof at pleasure within the Concavity: Then take some other convenient Vessel, and put pure Spring∣water therein; then having filled the Vessel to a known Measure, make a Mark precisely round the very edge of the Water, then take the solid body and put it therein, then take out as much of the Water (as by means of the body put therein) is arisen above the Mark, untill the Water do justly touch at the Mark again: then put the Water taken forth into the

Hollow Cube, and find the solid Content thereof (being transformed into a Cubick Body) in Feet, Inches, and parts of an Inch: Which Con∣tent is the just solidity of the Body put into the Water. (Archimedes by this Proposition found the deceit of the Crown of Gold which Gelo the Son of Hiero had vowed unto his Gods: now the Workmen had mixed Silver with the Gold, which Theft was disco∣vered by the great skill of Archimedes)* 1.2 And herein you must be very curious not to spill any of the Water, or take out of the Vessel, or put into the Hollow Cube, any more than the just quantity arisen above the Mark, for if you do it will produce infinite Errours, and thus may the Solidity of any Irregular Body be found.

Notes

• * 1.1

  Whose Surface is bounded by a Line called by Proclus a Helicoides, but it may also be called a Helix, a Twist or Wreath, &c.
• * 1.2

  See Procl. lib. 2. cap. 3. & Viturvius lib. 9. cap 3.