147:
295:. Each arm of the sector was marked with four lines on the front, and three on the back, and the pivot had a dimple that would accept the point of a divider. The lines and scales on each arm where identical, and arranged in the same order as you moved from the inner edge to the outer edge, thus forming seven pairs of lines. All the calculations could be performed with some combination of five very simple steps: measuring some length, separation or object width with the divider; opening the arms of the sector and setting the crosswise distance between two corresponding points on a pair of lines to the divider separation; measuring the crosswise distance between two corresponding points on a pair of lines once the sector had been set to some separation; reading a value from one of the scales at a point where the crosswise distances matches a divider separation; and reading a value off a scale where the distance from the pivot matches a divider. Galileo did not describe how the scales where constructed, he considered that a trade secret, but the details can be inferred. Scale markings were placed with an accuracy of about 1%.
177:
1703:
measure the diameter of the cannonball with the known charge and set the sector crosswise at this cannonball's material mark on the metallic lines to that diameter. The crosswise distance at the second cannonball's material type gives you the diameter of a cannonball in this material that is the same weight as the first ball. We need to scale this length down stereometrically to the given diameter of the second ball to get the correct charge, so we set the crosswise distance on the stereometric lines at 100–100 to the crosswise distance we just measured from the metallic lines, and then see where the crosswise distance on the stereometric lines matches the actual diameter of the second ball. The charge required is then in the ratio of this scale reading to 100 compared to the ball with known charge. You could then use the arithmetic lines to scale the charge weight in this ratio.
247:
239:
1436:
then drop the last three digits from our number, and if the number we dropped was more than 500, we add one to the remainder. We measure the crosswise distance on the stereometric lines at the remainder value, and place this against the arithmetic lines to find the cube root. The largest number that can be handled without rescaling here is 148,000. For “large” numbers we set the sector crosswise at 100–100 on the stereometric lines to the distance from the pivot to the point 100 on the arithmetic lines, and instead of dropping three digits, we drop four. This can handle numbers from 10,000 up to 1,480,000 without rescaling. For practical use, you should use the medium number procedure for all values up to 148,000 that are not within about 2% of a multiple of 10,000.
162:
603:
lines to the divider separation, any number will do, say 20. Then measure the length of the corresponding side in each of the other figures, and read the
Geometric Line scale value where the crosswise distance matches these lengths. Add together all the scale readings, including the 20 we originally set. At the combined value on the geometric lines, measure the crosswise distance. This will be the length of the side of the figure that has the combined area of the set. You can then use the arithmetic scale to scale all the other side lengths in the largest figure to match. This procedure will work for any closed figure made from straight lines.
50:
34:
477:
on the arithmetic lines to the distance just measured to P0. If the interest rate for the period is say 6%, then set the divider to the crosswise distance at 106-106. Place the divider at the pivot, and see where the other end falls on the arithmetic lines. This is the value of the investment at the end of the first period. Now set the crosswise distance at 100-100 again to the current divider separation and repeat the procedure for as many periods as needed.
1712:
2665:
formula which we have to infer as
Galileo does not describe how this scale was constructed. The name of these lines derives from the fact that they were added by Galileo to an earlier version of his sector. These lines are used for squaring circular segments, that is finding the side length of a square that is equal in area to a circular segment with a given chord length and height, where the segment is at most a semicircle.
607:
geometric lines to this distance. Next take your number and divide by 100, rounding to the nearest integer. So for example 8679 becomes 87. If this number is greater than 50 (the largest value on the geometric lines scale) then it must be reduced, in this example perhaps divided by 3 to make 29. Next measure the crosswise distance on the geometric lines at 29, this distance on the arithmetic lines represents
190:(1532 – ca 1608) was an Italian mathematician who is best known for his invention of the "proportional eight-pointed compass" which has two arms with cursors that allow the solution of problems in measuring the circumference, area and angles of a circle. In 1567 he published a single sheet treatise in Venice showing illustrations of his device. In 1585
271:. Galileo provided Mazzoleni and his family with room and board, and paid him two-thirds of the 35 lire selling price; Galileo would charge 120 lire for a course teaching the use of the instrument, about half the annual wage of a skilled craftsmen. Most of his customers were wealthy noblemen, including
594:
Galileo describes how to use these lines to scale a figure such that the new figure has a given area ratio to the original, how to measure the area ratio of two similar figures, how to combine a set of similar figures into another similar figure such that the resulting figure has the combined area of
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is the distance from the pivot to the zero on the outer scale, and the zero is marked with a small square. The outer scale zero lies close to the point marked 6 on the inner scale. The inner scale at first glance also appears linear, but its point spacings are actually determined by a fairly complex
715:
The procedure for calculating the square root of a “small” number, a number “around 100”, is simpler: we don't bother dividing by 100 at the beginning but otherwise perform the same procedure. At the end, divide the resulting square root estimate by 10. For "large" numbers ("around 50,000"), set the
476:
As an example, the procedure for calculating the compounded value of an investment is as follows. If the initial investment is P0, set the divider to the distance from the pivot to the point marked at P0 on the arithmetic lines. Open the instrument and set the crosswise distance at the point 100–100
1702:
These lines were of interest to artillerymen to solve the problem of “making the caliber”, that is how to figure out the correct powder charge to use for a cannonball of some size and material, when the correct charge is known for a cannonball of a different size and material. To do that, you would
606:
The procedure for calculating a square root varies depending on the size of the radicand. For a "medium" number ("in the region of 5,000"), start by measuring the distance from the pivot to the point marked 40 on the arithmetic lines, and setting the crosswise distance of the sector at 16–16 on the
1435:
The procedure for calculating cube roots is like that used for square roots, except that it only works for values of 1,000 or more. For “medium” numbers we set the sector crosswise at 64–64 on the stereometric lines to the distance from the pivot to the point marked 40 on the arithmetic lines. We
853:
Galileo describes how to use these lines to find the corresponding side length in a similar solid where the solid has a given volume ratio to the original, how to determine the volume ratio of two similar solids given the lengths of a pair of corresponding sides, how to find the side lengths of a
290:
Galileo described how to perform 32 different calculations with the sector in his 1606 manual. In the introduction, Galileo wrote that his intention in producing the sector was to enable people who had not studied mathematics to perform complex calculations without having to know the mathematical
602:
As an example, the procedure for producing a similar figure that has the combined area of a set of similar figures, is as follows: Choose a side in the largest figure and measure its length with a divider. Open the sector and set the crosswise distance at some intermediate value on the geometric
81:
in use from the end of the sixteenth century until the nineteenth century. It is an instrument consisting of two rulers of equal length joined by a hinge. A number of scales are inscribed upon the instrument which facilitate various mathematical calculations. It was used for solving problems in
1932:
254:
Galileo first developed his sector in the early 1590s as a tool for artillerymen. By 1597 it had evolved into an instrument that had much broader utility. It could be used, for example, to calculate the area of any plane figure constructed from a combination of straight lines and semi-circles.
595:
the set, how to construct a similar figure that has area equal to the difference in area of two other similar figures, how to find the square root of a number, how to arrange N soldiers into a grid where the ratio of rows to columns is some specified value, and how to find the
214:'s 1598 book. The sector Hood described was intended for use as a surveying instrument and included sights and a mounting socket for attaching the instrument to a pole or post, as well as an arc scale and an additional sliding leg. In the 1600s, the British mathematician
1719:
sides inscribed in a given circle. In
Galileo's sector design, he inverted this scale so that the numbers increase as they go away from the hinge and are more uniformly spaced. Later designs both in England and Continental Europe reverted to the original polygon
2595:. This zero point is about 70% of the way out along the arm. The inner scale is also described to run from 18 down to 0, but the sector in the Galileo Museum is only marked from 17. The zero point on the inner scale lies further out on the arm, at a distance of
2835:), that the area of the segment is the difference between the area of the pie slice defined by where the chord cuts the circle, and the triangle formed by the chord and the two radii that touch the ends of the chord. The base of the triangle has length
661:
We can choose any convenient value, e.g. 10, setting the sector crosswise distance at 10 to the divider separation, and then measure the crosswise distance at 30 on the geometric lines, then place the divider against the arithmetic lines to measure
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parts. The procedure for finding the radius of the enclosing circle is as follows: Open the sector and set the crosswise distance at the point 6–6 on the polygraphic lines to the desired side length. The distance measured crosswise at
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The outermost set of lines on the back have a double scale, an outer and an inner scale. The outer scale is linear and runs from 18 down to 0 as you move away from the pivot, and the zero point is marked with a ⌓, the symbol for a
468:
Galileo describes how to use these scales to divide a line into a number of equal parts, how to measure any fraction of a line, how to produce a scaled version of a figure or map, how to solve Euclid's Golden Rule (also called the
2470:. There is, of course, no such polygon, but this gives us a reference point on the Tetragonic Lines, the indicated circle, where it is easy to read off crosswise the radius of the circle that is equal in area to the polygon with
716:
sector crosswise at 10–10 on the geometric lines to the distance from the pivot to the point at 100 on the arithmetic lines. Divide the number by 1000 and round to the nearest integer. Then follow a similar procedure as before.
1694:
3791:, with applications in surveying and ballistics. The sector could also be used to easily determine the elevation of a cannon by inserting one arm into the barrel and reading the elevation from the location of the plumb bob.
2213:
1818:
589:
845:
209:
The two earliest known sectors in
England were made by Robert Beckit and Charles Whitwell, respectively, both dated 1597. These have a strong resemblance to the description of the device given by English mathematician
267:. It took him more than a year to solve this problem. The instrument we know today as Galileo's sector is the version with this added capability that he began to produce in 1599 with the help of the instrument maker
463:
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2422:
2004:
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added further scales in the 1590s, and published a book on the subject in 1606. Galileo's sector was first designed for military applications, but evolved into a general purpose calculating tool.
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41:
with a brass hinge. This side has scales for lines of lines (L), secants (S), chords (C), and polygons (POL), along with a scale of 10ths of inches on the outer edges forming a straight 12-inch
146:
3450:
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2082:
Galileo describes how to use these lines to find the radius of an enclosing circle for a polygon of n sides of a given length or in the other direction how to find the length of a
1806:
3001:
2635:
1728:, innermost scale on the back of the instrument, is labelled from 3 to 15, and the distance from the pivot is inversely proportional to the side length of a regular polygon of
854:
similar solid that has the combined volume of a set of other similar solids, how to find the cube root of a number, how to find the two values intermediate between two numbers
165:
1179:
1143:
687:
632:
3252:
711:
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1430:
1276:
971:
659:
1393:
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3158:
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1007:
928:
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389:
154:
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3196:
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359:
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1610:
1341:
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739:, the geometry of three-dimensional objects. The scale is marked to 148, and the distance from the pivot is proportional to the cube root. If we call the length
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275:, to whom Galileo sold a sector made of silver. More than a hundred were made in all, but only three are known to exist today: one in the Putnam Gallery at
719:
Galileo provides no further guidance, or refinement. Knowing which procedure to use for a given number requires some thought, and an appreciation for the
176:
1617:
198:'s hypothesis on the incommensurability of infinitesimals, thus confirming the existence of the "minimum" which laid the basis of his own atomic theory.
4177:
1927:{\displaystyle {\frac {P_{1}}{P_{2}}}={\frac {\operatorname {crd} {\dfrac {360^{\circ }}{n_{2}}}}{\operatorname {crd} {\dfrac {360^{\circ }}{n_{1}}}}}.}
2142:
522:
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is the distance from the pivot to the point marked 3. There is a circle on the scale that lies nearly midway between 6 and 7. The name comes from
4386:
768:
489:, which have a scale numbered from 1 to 50, with lengths proportional to the square root, called geometric because they are used for finding the
1363:
on the arithmetic lines, and sees at what value on the stereometric lines this distance fits crosswise, thus multiplying the previous result by
4187:
184:
The sector was invented, essentially simultaneously and independently, by a number of different people prior to the start of the 17th century.
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3787:
which, when in place, would lock the arms at 90° to each other. The sector could then be used for sighting and distance measurements using
2828:. The crosswise distance between the points n-n on the inner scale is the side length of the square equal in area to the circular segment.
473:), how to convert a value in one currency into the value in another currency, and how to calculate the compounded value of an investment.
2250:
of regular polygons, that is, finding the side of a square whose area is the same as the given regular polygon. They can also be used to
1699:
The ratio of lengths on this scale is proportional to the ratio of diameters of two balls of the same weight but different materials.
4239:
397:
2280:
3590:
4096:"New theories for new instruments: Fabrizio Mordente's proportional compass and the genesis of Giordano Bruno's atomist geometry"
2688:. At the chord midpoint, measure the length of the line perpendicular to the chord to where it intersects the circle, the height
2370:
1947:
3050:
4356:
4155:
4067:
238:
53:
The other side of the same sector, with scales for a line of sines (S) and two lines of tangents (T), along with logarithmic
45:
when the sector is fully opened, and a scale of 100ths of a foot marked along the side (only barely visible in this picture).
311:, that is, a linear scale. The sector in the Galileo Museum is marked from 16 to 260. If we call the length from the pivot
2582:. It is just as easy to find the required side lengths for any two polygons of equal area with different number of sides.
3257:
246:
26:"Proportional compass" redirects here. For the adjustable dividers also sometimes called a "proportional compass", see
255:
Galileo was determined to improve his sector so that it could be used to calculate the area of any shape discussed in
202:
developed a "polymetric compass" c. 1670, including a scale for constructing regular polygons. The
Italian astronomer
850:
These lines operate in an analogous way to the geometric lines, except that they deal with volumes instead of areas.
226:, privately distributing a Latin manuscript explaining its construction and use. Gunter published this in English as
2139:
are marked from 13 down to 3 as you move away from the pivot, and the distance from the pivot can be inferred to be
272:
3519:, we set up a pair of similar triangles that share the angle made by the arms of the sector at the pivot, so that
4629:
4576:
3161:
3809:
3371:
161:
4232:
83:
1715:
The scale of polygons on a typical sector has lines proportional to the side length of a regular polygon of
4348:
4167:
1791:
4634:
4485:
3479:
is the distance from the pivot to the zero point on the outer scale. When we set the sector crosswise to
470:
211:
2943:
1748:
sides inscribed in a given circle, or directly proportional to the circumradius of a regular polygon of
4521:
4516:
2708:. Set the sector crosswise on the added lines at the zero of the outer scale to the half-chord length,
2598:
720:
2510:
on the
Tetragonic Lines crosswise to the polygon side length. Squaring the circle is then just using
1148:
1145:
Galileo's method is to first use the geometric lines to find the geometric mean of two of the sides,
1102:
665:
610:
268:
3201:
692:
4624:
4225:
2247:
1221:
on the stereometric lines, calibrating the sector so that the distance from the pivot to the point
2891:
1532:
or densities, with distance proportional to the inverse cube root. Given two materials of density
1398:
1244:
933:
637:
4592:
4480:
4287:
4272:
4262:
3771:
on the inner scale will be the side length of the square with area equal to that of the segment.
2668:
The procedure for square a circular segment is as follows. Measure the half-length of the chord,
1366:
78:
3522:
1562:
391:
the ratios of their lengths are in proportion to the ratios of the numbers. In modern notation:
4363:
4042:
3987:
3866:
1535:
308:
292:
91:
3694:
3143:
2447:
976:
897:
4584:
3784:
3026:
3006:
2032:
131:
4008:
364:
4527:
4509:
4465:
4418:
3455:
3167:
3003:. The area of the pie slice is the fraction of the area of the circle covered by the angle
2640:
2218:
2072:
337:
199:
242:
Galileo's geometrical and military compass, thought to have been made c. 1604 by
Mazzoleni
8:
4568:
4560:
4497:
4370:
4340:
4277:
3944:
2937:
2861:
2559:
2513:
2251:
1201:
using a divider, and then sets the sector crosswise to this distance at the point marked
223:
3499:
at the zero point and find the point on the outer scale where the crosswise distance is
2838:
1592:
1326:
1303:
1281:
1059:
742:
496:
314:
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3734:
3570:
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3482:
3351:
2885:
2811:
2791:
2771:
2751:
2731:
2711:
2691:
2671:
2539:
2493:
2473:
2427:
2350:
2260:
2110:
2089:
2012:
1771:
1751:
1731:
1346:
1224:
1204:
1184:
1082:
1039:
1030:
1012:
877:
857:
280:
276:
250:
Figure showing the scales of
Galileo's military compass, from his manual on the device.
134:, and sometimes a clamp at the end of one leg which allowed the device to be used as a
4073:
Meskens, A. (1997). "Michel
Coignet's contribution to the development of the sector".
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187:
127:
54:
27:
20:
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at Red Rose
Reproductions, including several videos demonstrating uses of the sector
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at which the radius of the circle is the same as the side length of the polygon, is
634:
Because our number was reduced to fit on the sector, we must scale the length up by
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2083:
1809:
1689:{\displaystyle {\frac {M_{1}}{M_{2}}}={\frac {\sqrt{\rho _{2}}}{\sqrt{\rho _{1}}}}}
291:
details involved. The sector was used in combination with a divider, also called a
264:
4544:
4504:
4297:
4248:
3946:
Traité de la construction et des principaux usages des instrumens de mathématique
2068:
1529:
203:
4111:
3905:
Scale details can be read from photographs presented on page 88 in Bennett, 2022
2367:
is the side length of the polygon. The radius of the circle with equal area is
222:
with divisions proportional to the spacing of latitudes along a meridian on the
4424:
4282:
4267:
4016:
2208:{\displaystyle L(n)=L_{t}{\sqrt {\frac {n}{{\sqrt {3}}\tan {\frac {180}{n}}}}}}
1525:
1448:, the outermost pair on the front face, are marked with the symbols "ORO" (for
736:
596:
490:
191:
122:. The sector derives its name from the fourth proposition of the sixth book of
87:
4205:
4199:
4140:
4086:
493:
and working with areas of plane figures. If we call the length from the pivot
4618:
4475:
4430:
4034:
3995:
3983:
3954:
Drake, Stillman (1976). "Galileo and the First Mechanical Computing Device".
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1181:
He then measures the distance along the arithmetic lines to the point marked
584:{\displaystyle {\frac {G_{1}}{G_{2}}}={\frac {\sqrt {n_{1}}}{\sqrt {n_{2}}}}}
284:
215:
4436:
4211:
4120:
4052:
The Description and Use of the Sector, Crosse-staffe, and Other Instruments
3923:
3919:
2831:
To see how this works, we start by noting (as can be seen in the figure in
2076:
840:{\displaystyle {\frac {S_{1}}{S_{2}}}={\frac {\sqrt{n_{1}}}{\sqrt{n_{2}}}}}
99:
19:
This article is about sector compasses. For sector mass spectrometers, see
3879:
4378:
4307:
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103:
49:
33:
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110:. Its several scales permitted easy and direct solutions of problems in
37:
A typical English sector, probably from the early 19th century, made of
3938:
263:. To do this, he needed to add the capability to calculate the area of
119:
4060:
Wissenschaftliche Instrumente in ihrer Zeit. Vom 15. – 19. Jahrhundert
3780:
195:
135:
130:
have their like sides proportional. Some sectors also incorporated a
115:
107:
1711:
218:
dispensed with accessories but added additional scales, including a
4586:
Galileo's Daughter: A Historical Memoir of Science, Faith, and Love
4302:
1029:, and how to find the side of a cube that has the same volume as a
95:
4217:
2536:. To square the polygon, all we do is set the sector crosswise at
1343:
He then measures the distance from the pivot to the point marked
2127:
on the polygraphic lines is the radius of the enclosing circle.
180:
Brass sector with dividers, probably made in Dresden around 1630
57:
for numbers (N), sines (S), and tangents (T) on the outer edges.
3929:
The Construction and Principal Uses of Mathematical Instruments
3044:
1515:
1485:
1475:
256:
123:
458:{\displaystyle {\frac {A_{1}}{A_{2}}}={\frac {n_{1}}{n_{2}}},}
4002:(ed.). (5th ed.). Francis Eglesfield. pp. 157–194.
3684:{\textstyle L_{inner}(n)=L_{a}{\sqrt {z^{2}\arcsin 1/z-z+x}}}
2340:{\displaystyle A(n)=L^{2}{\frac {n}{4\tan {\frac {180}{n}}}}}
42:
38:
2808:. Move to the point on the inner scale that is also marked
102:, and for computing various mathematical functions, such as
3751:
defined as before, then the crosswise distance measured at
2246:(quadrilateral), as the main purpose of these lines is the
1938:
1495:
1465:
1455:
151:
De fabrica et usu menti ad omnia horarum genera describenda
2417:{\displaystyle r=L{\frac {n}{4\pi \tan {\frac {180}{n}}}}}
2067:
These lines can be used to aid in the construction of any
1999:{\displaystyle P(n)={\frac {R}{2\sin {\dfrac {\pi }{n}}}}}
1278:
the side of a cube with the volume of a cuboid with sides
1505:
111:
3846:
3844:
3133:{\displaystyle A_{pie}=\theta /2r^{2}=r^{2}\arcsin(c/r)}
16:
Mathematical instrument consisting of two hinged rulers
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2601:
303:
The innermost scales of the instrument are called the
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4029:(in Italian) (3rd ed.). Padua: Paolo Frambotto.
3941:, 1723. Revised and expanded English translation of
4047:(5th ed.). Francis Eglesfield. pp. 1–156.
3949:(in French) (Revised ed.). P. Husson & al.
3338:{\displaystyle A_{seg}=c^{2}(z^{2}\arcsin 1/z-z+x)}
1812:length of a circular arc measured in degrees, then
4200:"Slide Rule and Sine Plate have a common ancestor"
4168:The Proportional Compass or Sector and its History
4150:, Jim Bennett, Sillabe Srl, Livorno, Italy, 2022.
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2656:
2629:
2574:
2548:
2528:
2502:
2482:
2462:
2436:
2416:
2359:
2339:
2269:
2234:
2207:
2119:
2098:
2059:
2021:
1998:
1926:
1800:
1780:
1760:
1740:
1688:
1604:
1581:
1551:
1424:
1387:
1355:
1335:
1315:
1293:
1270:
1233:
1213:
1193:
1173:
1137:
1091:
1071:
1048:
1021:
1001:
965:
922:
886:
866:
839:
754:
705:
681:
653:
626:
583:
508:
457:
383:
353:
326:
4026:Le Operazioni del Compasso Geometrico et Militare
3807:
1937:Using functional notation in terms of the modern
279:, one in the Museum of Decorative Art in Milan's
4616:
4011:Operations of the Geometric and Military Compass
3348:The distance from the pivot to the point marked
2086:that divides the circumference of a circle into
4387:Dialogue Concerning the Two Chief World Systems
3867:"Of the Use of the Meridian Line in Navigation"
3833:
3254:, then we can write the area of the segment as
4148:Catalogue of Surveying and Related Instruments
4118:
4100:Studies in History and Philosophy of Science A
4062:. Verlag der Buchhandlung Walther König 2010,
4233:
3880:Labor, Productivity, Wages in Italy 1270-1913
2556:to the side length, and measure crosswise at
1519:
1509:
1499:
1489:
1479:
1469:
1459:
1449:
4019:. Washington, D.C.: Smithsonian Institution.
3885:Towards a Global History of Prices and Wages
3883:, Paolo Malanima, conference proceedings of
1528:). These symbols are arranged by decreasing
4125:"The Sector: Its History, Scales, and Uses"
1788:is the length on the polygraphic scale and
4240:
4226:
3445:{\displaystyle L_{outer}(n)=L_{a}(1-n/20)}
4206:"Cabinetmaker's Sector Tour and Tutorial"
3918:
726:
157:is among the first to describe the sector
1710:
1706:
245:
237:
175:
160:
145:
48:
32:
4208:by Brendan Bernhardt Gaffney on YouTube
4129:IEEE Annals of the History of Computing
4093:
4072:
4022:
4006:
3850:
2130:
298:
4617:
4033:
3982:
3862:
3836:Figuratio Aristotelici Physici auditus
3567:. If we set the distance of the point
1036:To cube a rectangular cuboid of sides
480:
4357:Letter to the Grand Duchess Christina
4221:
3953:
3587:from the pivot on the inner scale to
2728:. Find the point on the outer scale,
1589:if we call the length from the pivot
1439:
1241:on the stereometric lines represents
735:are so called because they relate to
485:The next set of lines are called the
4182:"The Geometric and Military Compass”
3942:
2858:, and the height of the triangle is
1801:{\displaystyle \operatorname {crd} }
4247:
3814:Biographical Dictionary of Italians
2257:The area of a regular polygon with
2029:is the circumradius for a hexagon,
233:
13:
4194:A typical sector and how to use it
3968:10.1038/scientificamerican0476-104
2996:{\displaystyle r=(c^{2}+h^{2})/2h}
2748:, where the crosswise distance is
2630:{\textstyle {\sqrt {\pi /2}}L_{a}}
2585:
194:used Mordente's compass to refute
14:
4646:
4178:The Scales of the Galilean Sector
4161:
334:then given two marks with values
4293:Leaning Tower of Pisa experiment
126:, where it is demonstrated that
4212:"Acer-Ferrous Toolworks Sector"
1174:{\displaystyle g={\sqrt {ab}}.}
1138:{\displaystyle s={\sqrt{abc}}.}
682:{\displaystyle {\sqrt {8700}},}
627:{\displaystyle {\sqrt {2900}}.}
4172:Kochi Arts & Science Space
3932:(2nd ed.). J. Richardson.
3899:
3890:
3872:
3827:
3801:
3622:
3616:
3439:
3419:
3403:
3397:
3332:
3290:
3247:{\displaystyle z=(1+x^{2})/2x}
3230:
3211:
3127:
3113:
2979:
2953:
2923:
2911:
2788:must be less than or equal to
2490:sides if we set the sector at
2293:
2287:
2155:
2149:
2045:
2039:
1960:
1954:
706:{\displaystyle {\sqrt {8679}}}
1:
4184:by G. Galilei (archived 2008)
3912:
3774:
1808:represents the trigonometric
170:Usage du compas de proportion
4349:Letter to Benedetto Castelli
2929:{\displaystyle A_{t}=c(r-h)}
1768:sides of a given length. If
1425:{\displaystyle {\sqrt{abc}}}
1271:{\displaystyle {\sqrt{ab}},}
966:{\displaystyle n_{2}=r^{2}p}
654:{\displaystyle {\sqrt {3}}.}
7:
4112:10.1016/j.shpsa.2018.10.004
1388:{\displaystyle {\sqrt{c}},}
1009:for a given scaling factor
10:
4651:
4522:Galileo National Telescope
4044:The Works of Edmund Gunter
4007:Galilei, Galileo (1978) .
3992:The Works of Edmund Gunter
3816:(in Italian), vol. 76
3808:Camerota, Filippo (2012),
3560:{\displaystyle h/c=1-n/20}
1582:{\displaystyle \rho _{2},}
721:propagation of uncertainty
166:Clément Cyriaque de Mangin
141:
25:
18:
4537:
4446:
4405:
4316:
4255:
4196:by Christopher J. Sangwin
4141:10.1109/MAHC.2003.1179877
4087:10.1080/00033799700200501
4023:Galilei, Galileo (1649).
1552:{\displaystyle \rho _{1}}
689:which is close enough to
4486:Galileo's objective lens
3834:Bruno, Giordano (1585),
3794:
3724:{\displaystyle x=1-n/20}
3153:{\displaystyle \arcsin }
2463:{\displaystyle n=6.5437}
1002:{\displaystyle q=r^{3}p}
923:{\displaystyle n_{1}=rp}
4517:Galileo Galilei Airport
4288:Galilean transformation
4263:Observational astronomy
4202:by IMSAI Guy on YouTube
4094:Rossini, Paolo (2019).
4021:English translation of
3943:Bion, Nicolas (1723) .
3779:The sector came with a
3164:function. If we define
3036:{\displaystyle \theta }
3016:{\displaystyle \theta }
2060:{\displaystyle P(6)=R.}
1033:(square-cornered box).
307:from their division in
155:Giovanni Paolo Gallucci
4630:Mechanical calculators
4364:Discourse on the Tides
4119:Williams, Michael R.;
3765:
3745:
3725:
3685:
3581:
3561:
3513:
3493:
3473:
3446:
3368:on the outer scale is
3362:
3339:
3248:
3192:
3154:
3134:
3037:
3017:
2997:
2930:
2878:
2852:
2822:
2802:
2782:
2762:
2742:
2722:
2702:
2682:
2658:
2631:
2576:
2550:
2530:
2504:
2484:
2464:
2438:
2418:
2361:
2341:
2271:
2236:
2209:
2121:
2100:
2061:
2023:
2000:
1928:
1802:
1782:
1762:
1742:
1721:
1690:
1606:
1583:
1553:
1520:
1510:
1500:
1490:
1480:
1470:
1460:
1450:
1426:
1389:
1357:
1337:
1317:
1295:
1272:
1235:
1215:
1195:
1175:
1139:
1093:
1073:
1050:
1023:
1003:
967:
924:
888:
868:
841:
756:
727:The stereometric lines
707:
683:
655:
628:
585:
510:
459:
385:
384:{\displaystyle n_{2},}
355:
328:
309:arithmetic progression
269:Marc'Antonio Mazzoleni
251:
243:
181:
173:
158:
79:calculating instrument
58:
46:
3766:
3746:
3726:
3686:
3582:
3562:
3514:
3494:
3474:
3472:{\displaystyle L_{a}}
3447:
3363:
3340:
3249:
3193:
3191:{\displaystyle x=h/c}
3155:
3135:
3038:
3018:
2998:
2931:
2879:
2853:
2823:
2803:
2783:
2763:
2743:
2723:
2703:
2683:
2659:
2657:{\displaystyle L_{a}}
2632:
2577:
2551:
2531:
2505:
2485:
2465:
2439:
2419:
2362:
2342:
2272:
2237:
2235:{\displaystyle L_{t}}
2210:
2122:
2101:
2062:
2024:
2001:
1929:
1803:
1783:
1763:
1743:
1714:
1707:The polygraphic lines
1691:
1607:
1584:
1554:
1427:
1390:
1358:
1338:
1318:
1296:
1273:
1236:
1216:
1196:
1176:
1140:
1099:amounts to computing
1094:
1074:
1051:
1024:
1004:
968:
925:
889:
869:
842:
757:
708:
684:
656:
629:
586:
511:
460:
386:
356:
354:{\displaystyle n_{1}}
329:
249:
241:
179:
164:
149:
52:
36:
4528:Astronomers Monument
4481:Galileo's telescopes
4419:Michelagnolo Galilei
4273:Galileo's escapement
4188:Cole Military Sector
3988:"The Sector Altered"
3810:"Mordente, Fabrizio"
3755:
3735:
3695:
3591:
3571:
3523:
3503:
3483:
3456:
3372:
3352:
3258:
3202:
3168:
3144:
3051:
3027:
3007:
2944:
2892:
2886:area of the triangle
2862:
2839:
2812:
2792:
2772:
2752:
2732:
2712:
2692:
2672:
2641:
2599:
2560:
2540:
2514:
2494:
2474:
2448:
2428:
2371:
2351:
2281:
2261:
2219:
2143:
2131:The tetragonic lines
2111:
2090:
2073:equilateral triangle
2033:
2013:
1948:
1819:
1792:
1772:
1752:
1732:
1618:
1593:
1563:
1536:
1399:
1367:
1347:
1327:
1304:
1282:
1245:
1225:
1205:
1185:
1149:
1103:
1083:
1060:
1040:
1013:
977:
934:
898:
878:
858:
769:
743:
693:
666:
638:
611:
523:
497:
398:
365:
338:
315:
299:The arithmetic lines
200:Guidobaldo del Monte
71:proportional compass
4498:Galileo thermometer
4371:Discourse on Comets
4341:Letters on Sunspots
4278:Galilean invariance
4190:at the IBM Archives
4049:First published in
3956:Scientific American
3935:1st English edition
2940:, we can show that
2877:{\displaystyle r-h}
2575:{\displaystyle n=4}
2529:{\displaystyle n=4}
481:The geometric lines
228:De Sectore et Radio
224:Mercator projection
65:, also known as a
4635:Italian inventions
4538:In popular culture
4493:Tribune of Galileo
4325:De motu antiquiora
3761:
3741:
3721:
3681:
3577:
3557:
3509:
3489:
3469:
3442:
3358:
3335:
3244:
3188:
3150:
3130:
3033:
3013:
2993:
2938:Pythogras' theorem
2926:
2874:
2851:{\displaystyle 2c}
2848:
2818:
2798:
2778:
2758:
2738:
2718:
2698:
2678:
2654:
2627:
2572:
2546:
2526:
2500:
2480:
2460:
2434:
2414:
2357:
2337:
2267:
2232:
2205:
2117:
2096:
2057:
2019:
1996:
1991:
1924:
1916:
1882:
1798:
1778:
1758:
1738:
1722:
1686:
1605:{\displaystyle M,}
1602:
1579:
1549:
1518:), and "PIE" (for
1440:The metallic lines
1422:
1385:
1353:
1336:{\displaystyle 1.}
1333:
1316:{\displaystyle b,}
1313:
1294:{\displaystyle a,}
1291:
1268:
1231:
1211:
1191:
1171:
1135:
1089:
1072:{\displaystyle b,}
1069:
1046:
1031:rectangular cuboid
1019:
999:
963:
920:
884:
864:
837:
755:{\displaystyle S,}
752:
733:stereometric lines
703:
679:
651:
624:
581:
509:{\displaystyle G,}
506:
455:
381:
351:
327:{\displaystyle A,}
324:
281:Castello Sforzesco
277:Harvard University
273:Archduke Ferdinand
252:
244:
182:
174:
159:
59:
47:
4612:
4611:
4466:Galileo's paradox
4461:Villa Il Gioiello
4256:Scientific career
4180:quotations from:
4156:978-88-3340-322-9
4075:Annals of Science
4068:978-3-86560-772-0
4039:Leybourn, William
4000:Leybourn, William
3783:and a detachable
3764:{\displaystyle n}
3744:{\displaystyle z}
3679:
3580:{\displaystyle n}
3512:{\displaystyle h}
3492:{\displaystyle c}
3361:{\displaystyle n}
2821:{\displaystyle n}
2801:{\displaystyle c}
2781:{\displaystyle h}
2761:{\displaystyle h}
2741:{\displaystyle n}
2721:{\displaystyle c}
2701:{\displaystyle h}
2681:{\displaystyle c}
2615:
2549:{\displaystyle n}
2503:{\displaystyle n}
2483:{\displaystyle n}
2437:{\displaystyle n}
2412:
2409:
2360:{\displaystyle L}
2335:
2332:
2270:{\displaystyle n}
2252:square the circle
2203:
2202:
2199:
2183:
2120:{\displaystyle n}
2099:{\displaystyle n}
2071:from the 3-sided
2022:{\displaystyle R}
1994:
1990:
1919:
1915:
1881:
1844:
1781:{\displaystyle P}
1761:{\displaystyle n}
1741:{\displaystyle n}
1726:polygraphic lines
1684:
1683:
1666:
1643:
1420:
1380:
1356:{\displaystyle c}
1263:
1234:{\displaystyle g}
1214:{\displaystyle g}
1194:{\displaystyle g}
1166:
1130:
1092:{\displaystyle c}
1049:{\displaystyle a}
1022:{\displaystyle r}
887:{\displaystyle q}
867:{\displaystyle p}
835:
834:
817:
794:
701:
674:
646:
619:
579:
578:
566:
548:
450:
423:
283:, and one in the
265:circular segments
188:Fabrizio Mordente
128:similar triangles
28:Reduction compass
21:Sector instrument
4642:
4578:Starry Messenger
4554:Lamp At Midnight
4455:And yet it moves
4413:Vincenzo Galilei
4395:Two New Sciences
4333:Sidereus Nuncius
4242:
4235:
4228:
4219:
4218:
4144:
4115:
4090:
4048:
4030:
4020:
4015:. Translated by
4014:
4003:
3979:
3950:
3933:
3906:
3903:
3897:
3894:
3888:
3876:
3870:
3860:
3854:
3848:
3839:
3838:
3831:
3825:
3824:
3823:
3821:
3805:
3770:
3768:
3767:
3762:
3750:
3748:
3747:
3742:
3730:
3728:
3727:
3722:
3717:
3690:
3688:
3687:
3682:
3680:
3663:
3649:
3648:
3639:
3637:
3636:
3615:
3614:
3586:
3584:
3583:
3578:
3566:
3564:
3563:
3558:
3553:
3533:
3518:
3516:
3515:
3510:
3498:
3496:
3495:
3490:
3478:
3476:
3475:
3470:
3468:
3467:
3451:
3449:
3448:
3443:
3435:
3418:
3417:
3396:
3395:
3367:
3365:
3364:
3359:
3344:
3342:
3341:
3336:
3316:
3302:
3301:
3289:
3288:
3276:
3275:
3253:
3251:
3250:
3245:
3237:
3229:
3228:
3197:
3195:
3194:
3189:
3184:
3159:
3157:
3156:
3151:
3139:
3137:
3136:
3131:
3123:
3106:
3105:
3093:
3092:
3080:
3069:
3068:
3042:
3040:
3039:
3034:
3022:
3020:
3019:
3014:
3002:
3000:
2999:
2994:
2986:
2978:
2977:
2965:
2964:
2935:
2933:
2932:
2927:
2904:
2903:
2883:
2881:
2880:
2875:
2857:
2855:
2854:
2849:
2833:circular segment
2827:
2825:
2824:
2819:
2807:
2805:
2804:
2799:
2787:
2785:
2784:
2779:
2767:
2765:
2764:
2759:
2747:
2745:
2744:
2739:
2727:
2725:
2724:
2719:
2707:
2705:
2704:
2699:
2687:
2685:
2684:
2679:
2663:
2661:
2660:
2655:
2653:
2652:
2636:
2634:
2633:
2628:
2626:
2625:
2616:
2611:
2603:
2593:circular segment
2581:
2579:
2578:
2573:
2555:
2553:
2552:
2547:
2535:
2533:
2532:
2527:
2509:
2507:
2506:
2501:
2489:
2487:
2486:
2481:
2469:
2467:
2466:
2461:
2443:
2441:
2440:
2435:
2423:
2421:
2420:
2415:
2413:
2411:
2410:
2402:
2384:
2366:
2364:
2363:
2358:
2346:
2344:
2343:
2338:
2336:
2334:
2333:
2325:
2310:
2308:
2307:
2276:
2274:
2273:
2268:
2241:
2239:
2238:
2233:
2231:
2230:
2214:
2212:
2211:
2206:
2204:
2201:
2200:
2192:
2184:
2179:
2173:
2172:
2170:
2169:
2137:tetragonic lines
2126:
2124:
2123:
2118:
2105:
2103:
2102:
2097:
2075:to the 15-sided
2066:
2064:
2063:
2058:
2028:
2026:
2025:
2020:
2005:
2003:
2002:
1997:
1995:
1993:
1992:
1983:
1967:
1933:
1931:
1930:
1925:
1920:
1918:
1917:
1914:
1913:
1904:
1903:
1894:
1884:
1883:
1880:
1879:
1870:
1869:
1860:
1850:
1845:
1843:
1842:
1833:
1832:
1823:
1807:
1805:
1804:
1799:
1787:
1785:
1784:
1779:
1767:
1765:
1764:
1759:
1747:
1745:
1744:
1739:
1695:
1693:
1692:
1687:
1685:
1682:
1677:
1676:
1667:
1665:
1660:
1659:
1650:
1649:
1644:
1642:
1641:
1632:
1631:
1622:
1611:
1609:
1608:
1603:
1588:
1586:
1585:
1580:
1575:
1574:
1558:
1556:
1555:
1550:
1548:
1547:
1530:specific weights
1523:
1513:
1503:
1493:
1483:
1473:
1463:
1453:
1431:
1429:
1428:
1423:
1421:
1419:
1414:
1403:
1394:
1392:
1391:
1386:
1381:
1379:
1371:
1362:
1360:
1359:
1354:
1342:
1340:
1339:
1334:
1322:
1320:
1319:
1314:
1300:
1298:
1297:
1292:
1277:
1275:
1274:
1269:
1264:
1262:
1257:
1249:
1240:
1238:
1237:
1232:
1220:
1218:
1217:
1212:
1200:
1198:
1197:
1192:
1180:
1178:
1177:
1172:
1167:
1159:
1144:
1142:
1141:
1136:
1131:
1129:
1124:
1113:
1098:
1096:
1095:
1090:
1078:
1076:
1075:
1070:
1055:
1053:
1052:
1047:
1028:
1026:
1025:
1020:
1008:
1006:
1005:
1000:
995:
994:
972:
970:
969:
964:
959:
958:
946:
945:
929:
927:
926:
921:
910:
909:
893:
891:
890:
885:
873:
871:
870:
865:
846:
844:
843:
838:
836:
833:
828:
827:
818:
816:
811:
810:
801:
800:
795:
793:
792:
783:
782:
773:
761:
759:
758:
753:
712:
710:
709:
704:
702:
697:
688:
686:
685:
680:
675:
670:
660:
658:
657:
652:
647:
642:
633:
631:
630:
625:
620:
615:
599:of two numbers.
590:
588:
587:
582:
580:
577:
576:
567:
565:
564:
555:
554:
549:
547:
546:
537:
536:
527:
515:
513:
512:
507:
464:
462:
461:
456:
451:
449:
448:
439:
438:
429:
424:
422:
421:
412:
411:
402:
390:
388:
387:
382:
377:
376:
360:
358:
357:
352:
350:
349:
333:
331:
330:
325:
305:arithmetic lines
234:Galileo's sector
75:military compass
4650:
4649:
4645:
4644:
4643:
4641:
4640:
4639:
4625:Galileo Galilei
4615:
4614:
4613:
4608:
4602:Galileo's Dream
4594:Galileo Galilei
4546:Life of Galileo
4533:
4505:Galileo project
4442:
4401:
4312:
4298:Phases of Venus
4251:
4249:Galileo Galilei
4246:
4164:
4017:Drake, Stillman
3915:
3910:
3909:
3904:
3900:
3895:
3891:
3877:
3873:
3861:
3857:
3849:
3842:
3832:
3828:
3819:
3817:
3806:
3802:
3797:
3777:
3756:
3753:
3752:
3736:
3733:
3732:
3713:
3696:
3693:
3692:
3659:
3644:
3640:
3638:
3632:
3628:
3598:
3594:
3592:
3589:
3588:
3572:
3569:
3568:
3549:
3529:
3524:
3521:
3520:
3504:
3501:
3500:
3484:
3481:
3480:
3463:
3459:
3457:
3454:
3453:
3431:
3413:
3409:
3379:
3375:
3373:
3370:
3369:
3353:
3350:
3349:
3312:
3297:
3293:
3284:
3280:
3265:
3261:
3259:
3256:
3255:
3233:
3224:
3220:
3203:
3200:
3199:
3180:
3169:
3166:
3165:
3145:
3142:
3141:
3119:
3101:
3097:
3088:
3084:
3076:
3058:
3054:
3052:
3049:
3048:
3047:, this area is
3028:
3025:
3024:
3008:
3005:
3004:
2982:
2973:
2969:
2960:
2956:
2945:
2942:
2941:
2899:
2895:
2893:
2890:
2889:
2863:
2860:
2859:
2840:
2837:
2836:
2813:
2810:
2809:
2793:
2790:
2789:
2773:
2770:
2769:
2753:
2750:
2749:
2733:
2730:
2729:
2713:
2710:
2709:
2693:
2690:
2689:
2673:
2670:
2669:
2648:
2644:
2642:
2639:
2638:
2621:
2617:
2607:
2602:
2600:
2597:
2596:
2588:
2586:The added lines
2561:
2558:
2557:
2541:
2538:
2537:
2515:
2512:
2511:
2495:
2492:
2491:
2475:
2472:
2471:
2449:
2446:
2445:
2429:
2426:
2425:
2424:. The value of
2401:
2388:
2383:
2372:
2369:
2368:
2352:
2349:
2348:
2324:
2314:
2309:
2303:
2299:
2282:
2279:
2278:
2262:
2259:
2258:
2226:
2222:
2220:
2217:
2216:
2191:
2178:
2177:
2171:
2165:
2161:
2144:
2141:
2140:
2133:
2112:
2109:
2108:
2091:
2088:
2087:
2069:regular polygon
2034:
2031:
2030:
2014:
2011:
2010:
1981:
1971:
1966:
1949:
1946:
1945:
1909:
1905:
1899:
1895:
1892:
1885:
1875:
1871:
1865:
1861:
1858:
1851:
1849:
1838:
1834:
1828:
1824:
1822:
1820:
1817:
1816:
1793:
1790:
1789:
1773:
1770:
1769:
1753:
1750:
1749:
1733:
1730:
1729:
1709:
1678:
1672:
1668:
1661:
1655:
1651:
1648:
1637:
1633:
1627:
1623:
1621:
1619:
1616:
1615:
1594:
1591:
1590:
1570:
1566:
1564:
1561:
1560:
1543:
1539:
1537:
1534:
1533:
1442:
1415:
1404:
1402:
1400:
1397:
1396:
1375:
1370:
1368:
1365:
1364:
1348:
1345:
1344:
1328:
1325:
1324:
1305:
1302:
1301:
1283:
1280:
1279:
1258:
1250:
1248:
1246:
1243:
1242:
1226:
1223:
1222:
1206:
1203:
1202:
1186:
1183:
1182:
1158:
1150:
1147:
1146:
1125:
1114:
1112:
1104:
1101:
1100:
1084:
1081:
1080:
1061:
1058:
1057:
1041:
1038:
1037:
1014:
1011:
1010:
990:
986:
978:
975:
974:
954:
950:
941:
937:
935:
932:
931:
905:
901:
899:
896:
895:
879:
876:
875:
859:
856:
855:
829:
823:
819:
812:
806:
802:
799:
788:
784:
778:
774:
772:
770:
767:
766:
744:
741:
740:
729:
696:
694:
691:
690:
669:
667:
664:
663:
641:
639:
636:
635:
614:
612:
609:
608:
572:
568:
560:
556:
553:
542:
538:
532:
528:
526:
524:
521:
520:
498:
495:
494:
487:geometric lines
483:
444:
440:
434:
430:
428:
417:
413:
407:
403:
401:
399:
396:
395:
372:
368:
366:
363:
362:
345:
341:
339:
336:
335:
316:
313:
312:
301:
236:
204:Galileo Galilei
144:
55:Gunter's scales
31:
24:
17:
12:
11:
5:
4648:
4638:
4637:
4632:
4627:
4610:
4609:
4607:
4606:
4598:
4590:
4582:
4574:
4566:
4558:
4550:
4541:
4539:
4535:
4534:
4532:
4531:
4524:
4519:
4514:
4513:
4512:
4501:
4500:
4495:
4490:
4489:
4488:
4483:
4473:
4468:
4463:
4458:
4450:
4448:
4444:
4443:
4441:
4440:
4434:
4428:
4425:Vincenzo Gamba
4422:
4416:
4409:
4407:
4403:
4402:
4400:
4399:
4391:
4383:
4375:
4367:
4360:
4353:
4345:
4337:
4329:
4320:
4318:
4314:
4313:
4311:
4310:
4305:
4300:
4295:
4290:
4285:
4283:Galilean moons
4280:
4275:
4270:
4268:Galileo affair
4265:
4259:
4257:
4253:
4252:
4245:
4244:
4237:
4230:
4222:
4216:
4215:
4209:
4203:
4197:
4191:
4185:
4175:
4163:
4162:External links
4160:
4159:
4158:
4145:
4116:
4091:
4081:(2): 143–160.
4070:
4056:
4035:Gunter, Edmund
4031:
4004:
3996:Gunter, Edmund
3984:Foster, Samuel
3980:
3962:(4): 104–113.
3951:
3914:
3911:
3908:
3907:
3898:
3889:
3871:
3855:
3853:, p. 146.
3840:
3826:
3799:
3798:
3796:
3793:
3776:
3773:
3760:
3740:
3720:
3716:
3712:
3709:
3706:
3703:
3700:
3678:
3675:
3672:
3669:
3666:
3662:
3658:
3655:
3652:
3647:
3643:
3635:
3631:
3627:
3624:
3621:
3618:
3613:
3610:
3607:
3604:
3601:
3597:
3576:
3556:
3552:
3548:
3545:
3542:
3539:
3536:
3532:
3528:
3508:
3488:
3466:
3462:
3441:
3438:
3434:
3430:
3427:
3424:
3421:
3416:
3412:
3408:
3405:
3402:
3399:
3394:
3391:
3388:
3385:
3382:
3378:
3357:
3334:
3331:
3328:
3325:
3322:
3319:
3315:
3311:
3308:
3305:
3300:
3296:
3292:
3287:
3283:
3279:
3274:
3271:
3268:
3264:
3243:
3240:
3236:
3232:
3227:
3223:
3219:
3216:
3213:
3210:
3207:
3187:
3183:
3179:
3176:
3173:
3149:
3129:
3126:
3122:
3118:
3115:
3112:
3109:
3104:
3100:
3096:
3091:
3087:
3083:
3079:
3075:
3072:
3067:
3064:
3061:
3057:
3032:
3012:
2992:
2989:
2985:
2981:
2976:
2972:
2968:
2963:
2959:
2955:
2952:
2949:
2925:
2922:
2919:
2916:
2913:
2910:
2907:
2902:
2898:
2873:
2870:
2867:
2847:
2844:
2817:
2797:
2777:
2757:
2737:
2717:
2697:
2677:
2651:
2647:
2624:
2620:
2614:
2610:
2606:
2587:
2584:
2571:
2568:
2565:
2545:
2525:
2522:
2519:
2499:
2479:
2459:
2456:
2453:
2433:
2408:
2405:
2400:
2397:
2394:
2391:
2387:
2382:
2379:
2376:
2356:
2331:
2328:
2323:
2320:
2317:
2313:
2306:
2302:
2298:
2295:
2292:
2289:
2286:
2266:
2229:
2225:
2198:
2195:
2190:
2187:
2182:
2176:
2168:
2164:
2160:
2157:
2154:
2151:
2148:
2132:
2129:
2116:
2095:
2056:
2053:
2050:
2047:
2044:
2041:
2038:
2018:
2007:
2006:
1989:
1986:
1980:
1977:
1974:
1970:
1965:
1962:
1959:
1956:
1953:
1935:
1934:
1923:
1912:
1908:
1902:
1898:
1891:
1888:
1878:
1874:
1868:
1864:
1857:
1854:
1848:
1841:
1837:
1831:
1827:
1797:
1777:
1757:
1737:
1708:
1705:
1697:
1696:
1681:
1675:
1671:
1664:
1658:
1654:
1647:
1640:
1636:
1630:
1626:
1601:
1598:
1578:
1573:
1569:
1546:
1542:
1446:metallic lines
1441:
1438:
1418:
1413:
1410:
1407:
1384:
1378:
1374:
1352:
1332:
1312:
1309:
1290:
1287:
1267:
1261:
1256:
1253:
1230:
1210:
1190:
1170:
1165:
1162:
1157:
1154:
1134:
1128:
1123:
1120:
1117:
1111:
1108:
1088:
1068:
1065:
1045:
1018:
998:
993:
989:
985:
982:
962:
957:
953:
949:
944:
940:
919:
916:
913:
908:
904:
883:
863:
848:
847:
832:
826:
822:
815:
809:
805:
798:
791:
787:
781:
777:
751:
748:
728:
725:
700:
678:
673:
650:
645:
623:
618:
597:geometric mean
592:
591:
575:
571:
563:
559:
552:
545:
541:
535:
531:
505:
502:
491:geometric mean
482:
479:
466:
465:
454:
447:
443:
437:
433:
427:
420:
416:
410:
406:
380:
375:
371:
348:
344:
323:
320:
300:
297:
285:Galileo Museum
235:
232:
192:Giordano Bruno
153:(1592), where
143:
140:
88:multiplication
77:, was a major
15:
9:
6:
4:
3:
2:
4647:
4636:
4633:
4631:
4628:
4626:
4623:
4622:
4620:
4605:
4603:
4599:
4597:
4595:
4591:
4589:
4587:
4583:
4581:
4579:
4575:
4573:
4571:
4567:
4565:
4563:
4559:
4557:
4555:
4551:
4549:
4547:
4543:
4542:
4540:
4536:
4530:
4529:
4525:
4523:
4520:
4518:
4515:
4511:
4508:
4507:
4506:
4503:
4502:
4499:
4496:
4494:
4491:
4487:
4484:
4482:
4479:
4478:
4477:
4476:Museo Galileo
4474:
4472:
4469:
4467:
4464:
4462:
4459:
4456:
4452:
4451:
4449:
4445:
4438:
4435:
4432:
4431:Maria Celeste
4429:
4426:
4423:
4420:
4417:
4414:
4411:
4410:
4408:
4404:
4397:
4396:
4392:
4389:
4388:
4384:
4381:
4380:
4376:
4373:
4372:
4368:
4365:
4361:
4358:
4354:
4351:
4350:
4346:
4343:
4342:
4338:
4335:
4334:
4330:
4327:
4326:
4322:
4321:
4319:
4315:
4309:
4306:
4304:
4301:
4299:
4296:
4294:
4291:
4289:
4286:
4284:
4281:
4279:
4276:
4274:
4271:
4269:
4266:
4264:
4261:
4260:
4258:
4254:
4250:
4243:
4238:
4236:
4231:
4229:
4224:
4223:
4220:
4213:
4210:
4207:
4204:
4201:
4198:
4195:
4192:
4189:
4186:
4183:
4179:
4176:
4173:
4169:
4166:
4165:
4157:
4153:
4149:
4146:
4142:
4138:
4134:
4130:
4126:
4122:
4121:Tomash, Erwin
4117:
4113:
4109:
4105:
4101:
4097:
4092:
4088:
4084:
4080:
4076:
4071:
4069:
4065:
4061:
4057:
4055:(1624, 1636).
4054:
4053:
4046:
4045:
4040:
4036:
4032:
4028:
4027:
4018:
4013:
4012:
4005:
4001:
3997:
3993:
3989:
3985:
3981:
3977:
3973:
3969:
3965:
3961:
3957:
3952:
3948:
3947:
3940:
3937:published by
3936:
3931:
3930:
3925:
3924:Stone, Edmund
3921:
3920:Bion, Nicolas
3917:
3916:
3902:
3893:
3886:
3882:
3881:
3875:
3869:, pp. 99–140.
3868:
3864:
3859:
3852:
3847:
3845:
3837:
3830:
3815:
3811:
3804:
3800:
3792:
3790:
3789:triangulation
3786:
3782:
3772:
3758:
3738:
3718:
3714:
3710:
3707:
3704:
3701:
3698:
3676:
3673:
3670:
3667:
3664:
3660:
3656:
3653:
3650:
3645:
3641:
3633:
3629:
3625:
3619:
3611:
3608:
3605:
3602:
3599:
3595:
3574:
3554:
3550:
3546:
3543:
3540:
3537:
3534:
3530:
3526:
3506:
3486:
3464:
3460:
3436:
3432:
3428:
3425:
3422:
3414:
3410:
3406:
3400:
3392:
3389:
3386:
3383:
3380:
3376:
3355:
3346:
3329:
3326:
3323:
3320:
3317:
3313:
3309:
3306:
3303:
3298:
3294:
3285:
3281:
3277:
3272:
3269:
3266:
3262:
3241:
3238:
3234:
3225:
3221:
3217:
3214:
3208:
3205:
3185:
3181:
3177:
3174:
3171:
3163:
3147:
3124:
3120:
3116:
3110:
3107:
3102:
3098:
3094:
3089:
3085:
3081:
3077:
3073:
3070:
3065:
3062:
3059:
3055:
3046:
3030:
3010:
2990:
2987:
2983:
2974:
2970:
2966:
2961:
2957:
2950:
2947:
2939:
2920:
2917:
2914:
2908:
2905:
2900:
2896:
2887:
2871:
2868:
2865:
2845:
2842:
2834:
2829:
2815:
2795:
2775:
2755:
2735:
2715:
2695:
2675:
2666:
2649:
2645:
2622:
2618:
2612:
2608:
2604:
2594:
2583:
2569:
2566:
2563:
2543:
2523:
2520:
2517:
2497:
2477:
2457:
2454:
2451:
2431:
2406:
2403:
2398:
2395:
2392:
2389:
2385:
2380:
2377:
2374:
2354:
2329:
2326:
2321:
2318:
2315:
2311:
2304:
2300:
2296:
2290:
2284:
2264:
2255:
2253:
2249:
2245:
2227:
2223:
2196:
2193:
2188:
2185:
2180:
2174:
2166:
2162:
2158:
2152:
2146:
2138:
2128:
2114:
2093:
2085:
2080:
2078:
2074:
2070:
2054:
2051:
2048:
2042:
2036:
2016:
1987:
1984:
1978:
1975:
1972:
1968:
1963:
1957:
1951:
1944:
1943:
1942:
1940:
1921:
1910:
1906:
1900:
1896:
1889:
1886:
1876:
1872:
1866:
1862:
1855:
1852:
1846:
1839:
1835:
1829:
1825:
1815:
1814:
1813:
1811:
1795:
1775:
1755:
1735:
1727:
1718:
1713:
1704:
1700:
1679:
1673:
1669:
1662:
1656:
1652:
1645:
1638:
1634:
1628:
1624:
1614:
1613:
1612:
1599:
1596:
1576:
1571:
1567:
1544:
1540:
1531:
1527:
1522:
1517:
1512:
1508:), "MA" (for
1507:
1502:
1498:), "ST" (for
1497:
1492:
1488:), "FE" (for
1487:
1482:
1478:), "RA" (for
1477:
1472:
1468:), "AR" (for
1467:
1462:
1457:
1452:
1447:
1437:
1433:
1416:
1411:
1408:
1405:
1395:resulting in
1382:
1376:
1372:
1350:
1330:
1310:
1307:
1288:
1285:
1265:
1259:
1254:
1251:
1228:
1208:
1188:
1168:
1163:
1160:
1155:
1152:
1132:
1126:
1121:
1118:
1115:
1109:
1106:
1086:
1066:
1063:
1043:
1034:
1032:
1016:
996:
991:
987:
983:
980:
960:
955:
951:
947:
942:
938:
917:
914:
911:
906:
902:
881:
861:
851:
830:
824:
820:
813:
807:
803:
796:
789:
785:
779:
775:
765:
764:
763:
749:
746:
738:
734:
724:
722:
717:
713:
698:
676:
671:
648:
643:
621:
616:
604:
600:
598:
573:
569:
561:
557:
550:
543:
539:
533:
529:
519:
518:
517:
503:
500:
492:
488:
478:
474:
472:
471:Rule of Three
452:
445:
441:
435:
431:
425:
418:
414:
408:
404:
394:
393:
392:
378:
373:
369:
346:
342:
321:
318:
310:
306:
296:
294:
288:
287:in Florence.
286:
282:
278:
274:
270:
266:
262:
258:
248:
240:
231:
229:
225:
221:
220:meridian line
217:
216:Edmund Gunter
213:
207:
205:
201:
197:
193:
189:
185:
178:
171:
167:
163:
156:
152:
148:
139:
138:'s quadrant.
137:
133:
129:
125:
121:
117:
113:
109:
105:
101:
97:
93:
89:
85:
80:
76:
72:
68:
64:
56:
51:
44:
40:
35:
29:
22:
4604:(2009 novel)
4601:
4596:(2002 opera)
4593:
4585:
4577:
4569:
4561:
4553:
4545:
4526:
4470:
4437:Marina Gamba
4393:
4385:
4377:
4369:
4347:
4339:
4331:
4323:
4181:
4171:
4147:
4135:(1): 34–47.
4132:
4128:
4103:
4099:
4078:
4074:
4059:
4058:Kern, Ralf,
4051:
4043:
4025:
4010:
3991:
3959:
3955:
3945:
3928:
3901:
3896:Galilei 1606
3892:
3884:
3878:
3874:
3858:
3851:Meskens 1997
3835:
3829:
3818:, retrieved
3813:
3803:
3778:
3347:
3162:inverse sine
2830:
2667:
2589:
2256:
2243:
2136:
2134:
2081:
2077:pentadecagon
2008:
1936:
1725:
1723:
1716:
1701:
1698:
1458:), PIO (for
1445:
1443:
1434:
1432:as desired.
1035:
852:
849:
732:
730:
718:
714:
605:
601:
593:
486:
484:
475:
467:
304:
302:
289:
260:
253:
227:
219:
208:
186:
183:
169:
150:
104:square roots
100:trigonometry
74:
70:
66:
62:
60:
4588:(1999 book)
4580:(1996 book)
4572:(1975 film)
4564:(1968 film)
4556:(1947 play)
4548:(1943 play)
4379:The Assayer
4328:(1589–1592)
4308:Thermoscope
3863:Gunter 1673
737:stereometry
212:Thomas Hood
67:sector rule
4619:Categories
4510:spacecraft
4439:(mistress)
4433:(daughter)
3939:John Senex
3913:References
3775:Other uses
2248:quadrature
1941:function,
894:such that
120:navigation
108:cube roots
84:proportion
4421:(brother)
4106:: 60–68.
3820:9 October
3781:plumb bob
3708:−
3668:−
3654:
3544:−
3426:−
3321:−
3307:
3111:
3074:θ
3031:θ
3011:θ
2918:−
2884:, so the
2869:−
2605:π
2399:
2393:π
2322:
2277:sides is
2189:
1985:π
1979:
1901:∘
1890:
1867:∘
1856:
1670:ρ
1653:ρ
1568:ρ
1541:ρ
230:in 1623.
196:Aristotle
116:surveying
4415:(father)
4366:" (1616)
4359:" (1615)
4303:Celatone
4123:(2003).
4037:(1673).
3986:(1673).
3976:24950332
3926:(1758).
3785:quadrant
3140:, where
2936:. Using
2347:, where
2244:tetragon
2215:, where
261:Elements
132:quadrant
96:geometry
92:division
4570:Galileo
4562:Galileo
4447:Related
4041:(ed.).
3691:, with
3160:is the
3045:radians
1471:argento
293:compass
142:History
112:gunnery
4471:Sector
4406:Family
4398:(1638)
4390:(1632)
4382:(1623)
4374:(1619)
4352:(1613)
4344:(1613)
4336:(1610)
4154:
4066:
3974:
3887:, 2004
3731:, and
3651:arcsin
3452:where
3304:arcsin
3198:, and
3148:arcsin
3108:arcsin
3023:. For
2637:where
2458:6.5437
2009:where
1720:scale.
1521:pietra
1516:marble
1501:stagno
1486:copper
1476:silver
1461:piombo
516:then:
257:Euclid
172:, 1637
136:gunner
124:Euclid
98:, and
63:sector
4427:(son)
4317:Works
3994:. By
3972:JSTOR
3795:Notes
2084:chord
1810:chord
1526:stone
1511:marmo
1491:ferro
762:then
73:, or
39:ivory
4152:ISBN
4064:ISBN
3822:2019
2135:The
1939:sine
1724:The
1559:and
1496:iron
1481:rame
1466:lead
1456:gold
1444:The
1323:and
1079:and
973:and
874:and
731:The
699:8679
672:8700
617:2900
361:and
118:and
106:and
90:and
61:The
43:rule
4137:doi
4108:doi
4083:doi
3964:doi
3960:234
3043:in
2888:is
2404:180
2396:tan
2327:180
2319:tan
2194:180
2186:tan
1976:sin
1897:360
1887:crd
1863:360
1853:crd
1796:crd
1506:tin
1451:oro
259:'s
4621::
4170:,
4133:25
4131:.
4127:.
4104:76
4102:.
4098:.
4079:54
4077:.
3998:.
3990:.
3970:.
3958:.
3922:;
3865:,
3843:^
3812:,
3719:20
3555:20
3437:20
3345:.
2768:;
2254:.
2079:.
1524:,
1514:,
1504:,
1494:,
1484:,
1474:,
1464:,
1454:,
1331:1.
1056:,
930:,
723:.
168:,
114:,
94:,
86:,
69:,
4457:"
4453:"
4362:"
4355:"
4241:e
4234:t
4227:v
4174:.
4143:.
4139::
4114:.
4110::
4089:.
4085::
3978:.
3966::
3759:n
3739:z
3715:/
3711:n
3705:1
3702:=
3699:x
3677:x
3674:+
3671:z
3665:z
3661:/
3657:1
3646:2
3642:z
3634:a
3630:L
3626:=
3623:)
3620:n
3617:(
3612:r
3609:e
3606:n
3603:n
3600:i
3596:L
3575:n
3551:/
3547:n
3541:1
3538:=
3535:c
3531:/
3527:h
3507:h
3487:c
3465:a
3461:L
3440:)
3433:/
3429:n
3423:1
3420:(
3415:a
3411:L
3407:=
3404:)
3401:n
3398:(
3393:r
3390:e
3387:t
3384:u
3381:o
3377:L
3356:n
3333:)
3330:x
3327:+
3324:z
3318:z
3314:/
3310:1
3299:2
3295:z
3291:(
3286:2
3282:c
3278:=
3273:g
3270:e
3267:s
3263:A
3242:x
3239:2
3235:/
3231:)
3226:2
3222:x
3218:+
3215:1
3212:(
3209:=
3206:z
3186:c
3182:/
3178:h
3175:=
3172:x
3128:)
3125:r
3121:/
3117:c
3114:(
3103:2
3099:r
3095:=
3090:2
3086:r
3082:2
3078:/
3071:=
3066:e
3063:i
3060:p
3056:A
2991:h
2988:2
2984:/
2980:)
2975:2
2971:h
2967:+
2962:2
2958:c
2954:(
2951:=
2948:r
2924:)
2921:h
2915:r
2912:(
2909:c
2906:=
2901:t
2897:A
2872:h
2866:r
2846:c
2843:2
2816:n
2796:c
2776:h
2756:h
2736:n
2716:c
2696:h
2676:c
2650:a
2646:L
2623:a
2619:L
2613:2
2609:/
2570:4
2567:=
2564:n
2544:n
2524:4
2521:=
2518:n
2498:n
2478:n
2455:=
2452:n
2432:n
2407:n
2390:4
2386:n
2381:L
2378:=
2375:r
2355:L
2330:n
2316:4
2312:n
2305:2
2301:L
2297:=
2294:)
2291:n
2288:(
2285:A
2265:n
2228:t
2224:L
2197:n
2181:3
2175:n
2167:t
2163:L
2159:=
2156:)
2153:n
2150:(
2147:L
2115:n
2094:n
2055:.
2052:R
2049:=
2046:)
2043:6
2040:(
2037:P
2017:R
1988:n
1973:2
1969:R
1964:=
1961:)
1958:n
1955:(
1952:P
1922:.
1911:1
1907:n
1877:2
1873:n
1847:=
1840:2
1836:P
1830:1
1826:P
1776:P
1756:n
1736:n
1717:n
1680:3
1674:1
1663:3
1657:2
1646:=
1639:2
1635:M
1629:1
1625:M
1600:,
1597:M
1577:,
1572:2
1545:1
1417:3
1412:c
1409:b
1406:a
1383:,
1377:3
1373:c
1351:c
1311:,
1308:b
1289:,
1286:a
1266:,
1260:3
1255:b
1252:a
1229:g
1209:g
1189:g
1169:.
1164:b
1161:a
1156:=
1153:g
1133:.
1127:3
1122:c
1119:b
1116:a
1110:=
1107:s
1087:c
1067:,
1064:b
1044:a
1017:r
997:p
992:3
988:r
984:=
981:q
961:p
956:2
952:r
948:=
943:2
939:n
918:p
915:r
912:=
907:1
903:n
882:q
862:p
831:3
825:2
821:n
814:3
808:1
804:n
797:=
790:2
786:S
780:1
776:S
750:,
747:S
677:,
649:.
644:3
622:.
574:2
570:n
562:1
558:n
551:=
544:2
540:G
534:1
530:G
504:,
501:G
453:,
446:2
442:n
436:1
432:n
426:=
419:2
415:A
409:1
405:A
379:,
374:2
370:n
347:1
343:n
322:,
319:A
30:.
23:.
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