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Sector Explained – A Revolution in Calculation

The ingenious sector device enabled a revolution in mathematics, science and technology during the Renaissance era. This comprehensive guide dives into the history, workings and immense impact of this groundbreaking calculating instrument.

The Mathematical Landscape Before the Sector

During the 15th and 16th centuries, mathematicians were in search of tools to mechanize complex calculations. Trigonometric tables had been developed for sine, tangent and secant functions. Logarithms had been discovered. But operations like long multiplication, division, square roots, cube roots, exponents and trig functions still required laborious manual methods.

A device was needed to accelerate these important mathematical operations that would enable advances in astronomy, navigation, architecture, warfare and more. The sector would meet this need and transform Renaissance science and technology.

Key Inventors Behind the Sector

The sector was independently conceived of at around 1554 by several mathematical pioneers across Europe:

Fabrizio Mordente (1532-1608) was an Italian mathematician who is considered the earliest inventor of the sector. Born in Salerno, he studied at the University of Naples and then embarked on extensive travels for over 10 years throughout Europe, the Middle East, India and beyond. He documented being inspired to create a “compass of proportion” in 1554, likely due to exposure to Arabic and Indian mathematics.

Fabrizio Mordente
Fabrizio Mordente, the earliest documented inventor of the sector. Credit: Wikimedia Commons.

Thomas Hood (1556–1620) was an English mathematician and physician who described the mathematical uses of the sector in his 1598 book “The Use of Both the Globes, the Sphere, and the Sector”. His design incorporated sights and mounting points for use as a surveying instrument.

Galileo Galilei (1564-1642), the famous Italian astronomer, created his own improved design for a military compass sector between 1595-1598. He manufactured over 100 copies and even trained students in their effective use.

Robert Beckit and Charles Whitwell independently crafted sector designs in 1597 in England. Their configurations closely resembled Hood‘s model.

The sector built upon the concept of similar triangles having proportional sides. By physically forming triangles and proportions with the hinged rulers, complex calculations could be carried out graphically with remarkable speed and accuracy.

Key Features and Scales of the Sector

While individual sectors had unique designs, most contained several common scales and markings:

  • Line scales – Primary scales showing divisions of equal line segments, used for basic measurements.

  • Arithmetic lines – Scales in arithmetical progression from 1 to 250, allowing multiplication and division via proportional parts.

  • Geometric lines – Scales in geometric progression based on square roots, facilitating quick determination of proportions.

Engraved brass sector by Benjamin Cole
Engraved brass sector by Benjamin Cole, London, 1765. Credit: Met Museum.
  • Stereometric lines – Used for calculations involving 3-dimensional shapes and solids.

  • Metallic lines – Converting between weights of various metals based on their densities.

  • Polygraphic lines – Finding circle radii to inscribe regular polygons.

  • Tetragonic lines – Squaring circles, triangles and other geometrical shapes.

Movable cursors allowed users to set up proportions between variables and read off the results from aligned scales. This mechanized the graphical solution of complex mathematics.

Step-by-Step Use of the Sector

Consider how the sector could be used to calculate a cube root, which was exceptionally tedious without it:

  1. Set the sector legs to span 1 unit on the line scale. This forms the base of the proportional triangle.

  2. On the Geometric scale, slide the cursor until the number marking the cube root is reached. For example, for the cube root of 8, cursor goes to 2.

  3. The aligned position of the cursor directly gives the cube root answer on the line scale. In this case, the cube root of 8 is 2.

Similar steps allow division, multiplication, trigonometry and other complex math. Even illiterate users could operate the sector via its graphical interface.

Historical Impact on Mathematics and Science

The sector dramatically accelerated mathematical work and enabled new discoveries and applications:

  • Navigation – Distance calculations at sea became easier, leading to improved navigation accuracy.

  • Architecture – Measuring angles and computing geometric volumes was simplified.

  • Gunnery – Quick targeting calculations increased artillery accuracy and range.

  • Surveying – Land mapping became easier with sectors to measure terrain areas and distances.

  • Astronomy – Sectors assisted astronomical observations and predictions. Kepler used a sector for calculating planetary motions.

Tycho Brahe with mural quadrant
Tycho Brahe pictured with large mural quadrant. Credit: Wikimedia Commons.

The sector opened up new frontiers in mathematics, science and technology by eliminating the need for slow manual calculations. Its importance is comparable to that of the first calculators and computers centuries later.

Famous Users of the Sector

Many prominent scientists and mathematicians utilized the sector in their revolutionary work:

  • Galileo – Used his sector design to confirm his experimental concepts in physics and astronomy.

  • Johannes Kepler – Leveraged the sector to deduce the laws of planetary motion.

  • Tycho Brahe – Equipped his astronomical observatory with large sector instruments.

  • Christopher Wren – Employed sectors for architectural works after studying at Oxford University.

  • Isaac Newton – Applied sectors in his physics and optics experiments at Cambridge.

These great minds took advantage of the sector’s potential to accelerate their research and mathematical calculations.

Collecting Antique Sectors Today

As a relic of Renaissance calculation and science, original sectors are prized collector’s items today. However, very few remain, as many were made from wood and simply deteriorated over time. Ivory sectors are also extremely rare due to ivory trade restrictions.

Antique brass French and Italian made sectors may still be found at high prices – expect to pay hundreds of dollars, depending on age, condition and engraving quality. Museums may also have some sectors in their mathematical instrument collections available for public viewing.

These artifacts provide a tangible connection to the revolutionary era when the sector paved the way for modern computation and technology.

The Sector‘s Enduring Legacy

While mostly obsolete today, the ingenious sector helped propel the pace of science and mathematics in its time. Its lasting significance is twofold:

Firstly, it mechanized complex calculations, freeing up intellectual capacity for new discoveries. Secondly, it showed how even everyday measurement tools could be innovatively combined with mathematics to create powerful results.

The sector was an important step on the journey towards calculators and computers. Its principles were built upon by later tools like the slide rule.

So while simply a hinged ruler, the sector represented a pivotal advance in the quest to automate thought. Its legacy drives today‘s technologies and digital world.

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