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Transversal (instrument making)
Transversal (instrument making)
Transversal (instrument making)
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Transversal

Transversals are a geometric construction on a scientific instrument to allow a graduation to be read to a finer degree of accuracy. Their use creates what is sometimes called a diagonal scale, an engineering measuring instrument which is composed of a set of parallel straight lines which are obliquely crossed by another set of straight lines. Diagonal scales are used to measure small fractions of the unit of measurement.[1]

Transversals have been replaced in modern times by vernier scales. This method is based on the Intercept theorem (also known as Thales's theorem).

History

[edit]

Transversals were used at a time when finely graduated instruments were difficult to make. They were found on instruments starting in the early 14th century, but the inventor is unknown. In 1342 Levi Ben Gerson introduced an instrument called Jacob's staff (apparently invented the previous century by Jacob Ben Makir) and described the method of the transversal scale applied to the mentioned instrument.[2][3]

Thomas Digges mistakenly attributed the discovery of the transversal scale to the navigator and explorer Richard Chancellor (cited by some authors as watchmaker and with other names, among them: Richard Chansler or Richard Kantzler).[4][5][6][7][8][9] Its use on astronomical instruments only began in the late 16th century. Tycho Brahe used them and did much to popularize the technique.[10][11] The technique began to die out once verniers became common in the late 18th century – over a century after Pierre Vernier introduced the technique.

In the interim between transversals and the vernier scale, the nonius system, developed by Pedro Nunes, was used. However, it was never in common use. Tycho also used nonius methods, but he appears to be the only prominent astronomer to do so.

Etymology

[edit]

Diagonal scale is derived from the Latin word Diagonalis. The Latin word was originally coined from the Greek word diagōnios where dia means "through" and gonios denotes "corners".[1][12]

Principle of a diagonal scale

[edit]

Diagonal scale follows the principle of similar triangles where a short length is divided into number of parts in which sides are proportional.[13] Divided into required number of equal parts

Linear transversals

[edit]
Linear transversals
Black - grid lines
Blue - transversals
Green - instrument's graduations
Red - cursor line

Linear transversals were used on linear graduations. A grid of lines was constructed immediately adjacent to the linear graduations. The lines extending above the graduations formed part of the grid. The number of lines perpendicular to the extended graduation lines in the grid was dependent on the degree of fineness the instrument maker wished to provide.

A grid of five lines would permit determination of the measure to one-fifth of a graduation's division. A ten-line grid would permit tenths to be measured. The distance between the lines is not critical as long as the distance is precisely uniform. Greater distances makes for greater accuracy.

As seen in the illustration on the right, once the grid was scribed, diagonals (transverse lines) were scribed from the uppermost corner of a column in the grid to the opposite lowest corner. This line intersects the cross lines in the grid in equal intervals. By using an indicator such as a cursor or alidade, or by measuring using a pair of dividers with points on the same horizontal grid line, the closest point where the transversal crosses the grid is determined. That indicates the fraction of the graduation for the measure.

In the illustration, the reading is indicated by the vertical red line. This could be the edge of an alidade or a similar device. Since the cursor crosses the transversal closest to the fourth grid line from the top, the reading (assuming the leftmost long graduation line is 0.0) is 0.54.

Application

[edit]

Diagonal scale is used in engineering to read lengths with higher accuracy as it represents a unit into three different multiple in metres, centimeters and millimeters.[14] Diagonal scale is an important part in Engineering drawings.[15]

Circular transversals

[edit]

Circular transversals perform the same function as the linear ones but for circular arcs. In this case, the construction of the grid is significantly more complicated. A rectangular grid will not work. A grid of radial lines and circumferential arcs must be created. In addition, a linear transverse line will not divide the radial grid into equal segments. Circular arc segments must be constructed as transversals to provide the correct proportions.

Tycho Brahe

[edit]
Tycho Brahe's transversal.

Tycho Brahe created a grid of transversal lines made with secants between two groups of arcs that form two graduated limbs. The secants are drawn by joining the division of a limb with the next division of the other limb, and so on (see figure with the magnification of 2 degrees of the Tycho Brahe's quadrant of 2m radius).[10]

He drew, for each degree, six straight transversals in an alternate mode forming a "V" and each transversal consisted of 9 points that divided it into 10 parts, which multiplied by 6 give 60 minutes.[16] While Abd al-Mun'im al 'Âmilî (16th century) drew them all in the same direction (although his instrument has less precision).[17]

Other authors

[edit]

The method of the "straight transversals" applied to the measurements of angles on circular or semicircular limbs in astronomical and geographic instruments was treated by several authors. Studying the accuracy of the system, some of them indicated the convenience of employing "Circular transversals", instead of the "straight transversals".[18]

See also

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References

[edit]

Bibliography

[edit]
[edit]
Revisions and contributorsEdit on WikipediaRead on Wikipedia

Transversal (instrument making)

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from Grokipedia
In instrument making, a transversal is a geometric applied to the scales of scientific instruments to enable more precise readings of graduations by magnifying subdivisions through the use of diagonal or slanted lines that intersect with primary markings. This technique allows measurements to be resolved to fractions of the main scale units, such as seconds of arc or finer increments, without requiring physically larger instruments. The transversal scale was pioneered in the early 14th century by the Jewish scholar and astronomer Levi ben Gerson (also known as ), who described its application in his astronomical treatise The Wars of the Lord, completed around 1329. Ben Gerson introduced the method to improve angular measurements on devices like astrolabes and the (a portable instrument for sighting celestial bodies), where it facilitated readings to one-fifteenth of a degree or better by aligning dividers or sights with intersecting transversal lines. His innovation addressed instrumental errors caused by the limitations of direct visual estimation on coarsely divided scales, marking a significant advancement in medieval precision instrumentation. Functionally, a transversal operates by creating a series of parallel slanted lines (often in a V-shaped or diagonal configuration) adjacent to a standard linear or circular scale; the distance between intersections with horizontal or radial lines corresponds to amplified subdivisions, enabling users to interpolate values more accurately using tools like compasses or sliding vanes. For instance, on a quadrant or , this allowed astronomers to measure small angular separations between stars with reduced errors. The design influenced later developments, including the nonius scale by in the and the in the 17th, and was employed in tools like cross-staffs as well as in surveying instruments. Transversals remained in use through the , notably by on his large quadrant in the late , underscoring their enduring role in enhancing observational accuracy before the widespread adoption of micrometers.

Fundamentals

Definition

In instrument making, a transversal is a geometric designed to enhance the precision of graduations on scientific instruments, allowing for measurements finer than those achievable with conventional linear scales. This technique employs a series of parallel lines intersected by oblique lines to form a diagonal network, where the positions of intersections correspond to subdivisions of the primary scale. By leveraging these diagonal alignments, transversals enable the reading of small fractional units, such as tenths, twentieths, or even smaller increments, without the need for densely packed engravings that would be difficult to produce or read. The core advantage of transversals lies in their distinction from direct linear divisions, which rely on uniform spacing along a straight line and are limited by the physical resolution of engraving tools and human eyesight. Instead, transversals magnify the effective scale through the of intersecting lines, effectively expanding the visual representation of minute divisions across a broader area. This approach draws on the principle of similar triangles to ensure proportional accuracy in readings. Introduced in the , transversals were developed specifically for precision tools such as the , a cross-staff used in astronomy and to measure angular distances with improved accuracy over earlier methods. Their application extended to various instruments requiring fine angular or linear measurements, marking a significant advancement in pre-modern .

Etymology

The term "transversal" derives from the Medieval Latin transversalis, meaning "running or lying across," which itself stems from the Latin transversus, the past of transvertere ("to turn across"), combining trans- ("across") with vertere ("to turn"). This root emphasizes a crossing or oblique orientation, aligning with its geometric usage as a line intersecting two or more others. In the context of instrument making, the term evolved to describe scales positioned at an to enhance precision, reflecting this inherent sense of traversal. Closely linked to "transversal" is the term "diagonal," originating from the Latin diagonālis, borrowed from diagōnios ("from to "), where dia- means "through" and gōnia denotes "" or "corner." In , a diagonal connects non-adjacent vertices of a , providing a conceptual bridge to the slanted lines used in early measuring devices. This geometric foundation facilitated the adoption of both terms in scientific instrumentation during the , where diagonal constructions allowed for subdivided readings beyond linear limits. The terminology in instrument making initially favored "diagonal scale," as proposed by Levi ben Gerson in the early for cross-staffs, but shifted toward "transversal" in 16th-century European texts. English mathematician , in his 1573 work Alae seu Scalae Mathematicae, prominently employed "transversal scale" to describe enhanced graduations on astronomical instruments, attributing its practical introduction to instrument maker Richard Chancellor. This evolution marked a broader integration of geometric lexicon into technical descriptions, standardizing "transversal" for angled scales in precision tools.

Principle of Diagonal Scales

The principle of diagonal scales in transversals rests on foundational , specifically the intercept theorem (also known as Thales' theorem), which asserts that a transversal line intersecting two or more divides each of those lines proportionally to the segments created on the transversal itself. This theorem, proven in Euclid's Elements (Book VI, Proposition 2), ensures that if parallel lines are cut by transversals, the segments intercepted on the transversals are proportional, providing a geometric basis for subdividing lengths accurately without direct measurement. This proportionality manifests through the formation of similar triangles when a transversal obliquely crosses a set of parallel lines spaced equally along a baseline. Consider a baseline of length LL divided by mm parallel lines into m+1m+1 equal segments, each of size L/(m+1)L/(m+1); a transversal intersecting these at an angle θ\theta creates a series of similar triangles where the ratio of corresponding sides equals the ratio of the intercepted segments on the transversal. The key relation is given by the similarity ratio: if the height perpendicular to the parallel lines is hh, then for a point at distance xx along the baseline, the corresponding segment on the transversal satisfies xL=sh\frac{x}{L} = \frac{s}{h}, where ss is the length along the transversal, allowing fractional readings scaled by the geometry. In block form for clarity: side of smaller triangleside of larger triangle=intercepted segment on transversalfull transversal length\frac{\text{side of smaller triangle}}{\text{side of larger triangle}} = \frac{\text{intercepted segment on transversal}}{\text{full transversal length}} For instance, to achieve divisions of 1/n1/n on a unit length, the transversal's intersections with the nn parallel lines enable direct proportional reading, as the oblique path ensures each intersection corresponds to L/nL/n. The oblique angle θ\theta of the transversal amplifies resolution by effectively compressing the apparent subdivision size on the instrument's scale; the smaller θ\theta, the greater the magnification factor, approximately 1/sin⁡θ1/\sin \theta, which stretches small intervals into more discernible physical separations without necessitating finer engravings. This geometric leverage arises because the slope of the transversal ( tan⁡θ\tan \theta ) projects horizontal divisions onto a vertical or perpendicular axis, multiplying the precision: a shallow angle allows a coarse baseline engraving to yield measurements finer than the engraving's limit, as the intersections spread out proportionally over the instrument's surface. Thus, diagonal scales convert proportional geometry into practical subdivision, bounded only by the angle's steepness and the number of parallel lines.

Historical Development

Origins and Early Uses

The transversal scale, a geometric method for enhancing measurement precision on scientific instruments, originated in the through the work of the Jewish astronomer and philosopher Levi ben Gerson (also known as ), who introduced it in Book V, Part 1, of his astronomical treatise The Wars of the Lord, completed around 1329. In this text, Levi described the application of a basic linear transversal scale on the (baculus Jacobi), an instrument designed primarily for measuring angular separations between celestial bodies, such as the altitude of stars or the sun, to aid in astronomical observations. This innovation allowed for more accurate readings by subdividing scale divisions using intersecting lines based on the principle of similar triangles, enabling finer graduations without enlarging the instrument itself. Levi's description represents the first documented use of transversals in Hebrew astronomical texts, where he emphasized their utility in overcoming limitations of direct scale readings for precise positional astronomy. The Jacob's staff with its transversal scale quickly found adoption in early 14th-century European contexts, particularly in and , for both astronomical purposes and emerging applications in and land surveying, such as determining heights of terrestrial objects or elevations at sea. An early physical example of this adoption appears on the Merton Astrolabe, crafted circa 1350 in , which features a simple transversal scale on its linear edges for improved angular measurements. By the , knowledge of transversals had spread to Latin astronomical works in , influencing instrument design in academic and practical settings. Basic linear forms of transversals were incorporated into medieval astrolabes, where they subdivided degree scales on the instrument's edges or plates for tasks like determining or stellar positions, and into quadrants, which used them along curved or straight edges to measure altitudes with greater resolution than plain graduations allowed. These early implementations prioritized conceptual simplicity, relying on the geometric proportionality of similar triangles to extend effective scale length, though detailed mathematical derivations were reserved for later treatises.

Tycho Brahe's Innovations

In the late 16th century, employed transversals extensively at his observatory on the island of Hven to conduct high-precision naked-eye observations of stellar positions, enabling unprecedented accuracy in mapping the heavens without telescopic aid. These instruments, constructed between 1576 and the early 1580s, supported Brahe's systematic cataloging of over 1,000 stars, with measurements of declinations and right ascensions that formed the foundation of his astronomical legacy. Brahe's key innovation involved adapting the transversal scale—originally a linear diagonal method—for circular instruments by creating a grid of transversal points between two concentric arcs to subdivide degrees into finer minutes and seconds. This technique used intersecting diagonal lines to divide each degree into 60 minutes, with further subdivisions achieved through dotted transversal lines across sections of ten minutes, allowing readings as precise as ten arcseconds on select instruments. Unlike earlier linear applications, Brahe's circular version minimized errors from , with deviations estimated at less than three arcseconds, thus enhancing resolution on large-scale arcs. This transversal design was prominently featured on Brahe's great globe, a approximately 1.5 meters in completed around 1580, where the equatorial and meridian circles were graduated using transversal points to distinguish single minutes and their subdivisions. Similarly, his mural quadrant, with a radius of about 1.94 meters and introduced in , incorporated the method to divide its arc into 90 degrees, each further subdivided so that minutes could be parsed into six parts, yielding ten arcseconds of readability. These features contributed to overall observational accuracies of 10 to 20 arcseconds in stellar altitude measurements. The completion of these instruments around 1580 marked a pinnacle in pre-telescopic astronomy, directly influencing the quality of Brahe's data that later informed Kepler's laws of planetary motion. By integrating transversals with refined sights and stable mountings at Uraniborg, Brahe achieved measurement precisions far surpassing contemporaries, typically within 15 to 35 arcseconds, revolutionizing empirical astronomy.

Later Authors and Decline

Following Tycho Brahe's innovations, subsequent astronomers built upon transversal designs to enhance precision in angular measurements, particularly for geodetic and observational purposes. (1580–1626), a Dutch and , significantly refined the application of circular transversals in surveys. In his seminal 1615–1616 measurements documented in Eratosthenes Batavus (published 1617), Snellius employed a large iron quadrant with an approximately 1.75-meter radius, equipped with transversal scales for reading minute angular divisions, constructed by instrument maker Blaeu. This instrument enabled the establishment of a meridian chain spanning approximately 130 kilometers between and , allowing Snellius to compute the with an accuracy of about 3.65% too small—a pioneering effort in practical that extended Tycho's foundational approaches to larger-scale networks. In the 17th and 18th centuries, transversals remained integral to instruments in prominent European observatories, where they facilitated high-resolution readings on large-scale devices. Johannes Hevelius (1611–1687), the Polish-Danish astronomer and mayor of , adapted and scaled up transversal-equipped quadrants for his rooftop observatory, the most advanced in at the time. Detailed in his Machinae Coelestis (1673–1679), Hevelius's mural and portable quadrants—some exceeding 3 meters in radius—incorporated transversal points to subdivide arcminutes into seconds, supporting meticulous lunar, planetary, and stellar observations without telescopic aids. These adaptations emphasized open-sight precision, with transversals aiding in the cataloging of over 1,500 stars and the mapping of lunar features, influencing observatory practices across institutions like those in and Greenwich. The decline of transversal scales accelerated in the late , supplanted by the nonius scale of (first described in 1542) and later the distinct popularized through Pierre Vernier's 1631 treatise La construction, l'usage et les effets du quadrant nouveau mathématique. Unlike transversals, which required diagonal or V-shaped subdivisions prone to visual estimation errors and complex alignment, the nonius and Vernier scales provided auxiliary methods for direct fractional readings, simplifying without added geometric intricacy. This ease of use, combined with greater portability for smaller instruments, rendered transversals obsolete in most astronomical and contexts by around 1800, though they lingered in some tools like theodolites and protractors into the early 19th century.

Types of Transversals

Linear Transversals

Linear transversals are constructed by etching a grid of parallel lines adjacent to the main scale graduations on a measuring rule, with these parallels intersected by a series of oblique diagonal lines to facilitate finer subdivisions. For dividing a primary unit into tenths, eleven equidistant parallel lines are drawn, spanning a height proportional to the desired magnification, while the main scale's divisions are projected vertically onto the top parallel and extended as perpendiculars through the others. Oblique transversals are then drawn connecting corresponding points across the grid, such as linking the end of the tenth division on the bottom parallel to the ninth on the top, creating a slanted network that distributes the subdivisions evenly. This configuration, often termed a V-type when the height is minimized by reusing parallels bidirectionally, adheres to the general principle of diagonal scales by leveraging geometric projection for enhanced readability. The reading method involves aligning a cursor, straightedge, or dividers with the relevant main graduation and observing the intersection of the oblique transversal with the parallel lines to determine the fractional value. For example, to measure 5.4 units on a scale marked in whole units, the cursor is set at the 5-unit mark, and the transversal line passing through this point will intersect the fourth parallel line in a ten-line grid, indicating 0.4 units since each parallel represents a tenth. The process requires estimating the intersection visually or with dividers spanning from a reference vertical to the diagonal crossing, ensuring the measurement's decimal place is read directly from the parallel's position without additional calculation. These scales are typically engraved on durable materials such as plates or wooden rules, with the lines finely incised to a spacing of less than 0.05 mm to prevent optical errors from misalignment or . examples, often 250 mm long and 40 mm wide with a 2 mm thickness, may be matt nickel-plated for resistance, while engravings are sometimes inked black to improve contrast under varying lighting. Ivory was historically used for lighter rules, but metal substrates dominate for precision due to their stability. The key advantage of linear transversals lies in providing a one-tenth resolution—or up to three significant digits—on scales with coarse primary markings, eliminating the need for micrometer screws or auxiliary devices while maintaining simplicity in fabrication. This geometric allows accurate fractional readings through visual alignment alone, making it suitable for instruments where space and limits constrain direct fine divisions.

Circular Transversals

Circular transversals represent an adaptation of the transversal method to curved surfaces, specifically for subdividing angular scales on circular instruments like quadrants and sextants. In this design, secant lines—diagonal connections—are drawn between corresponding points on inner and outer concentric arcs, creating a grid-like pattern that facilitates precise of fractions of degrees. For instance, this grid can divide a single degree into 60 minutes by aligning the or index with intersection points in the network, allowing readings without engraving minute divisions directly on the primary arc. The geometric adaptation maintains the core of similar triangles but accommodates by using radial lines as equivalents to straight parallels. These radial lines emanate from the instrument's , ensuring that the secant lines intersect them proportionally across the arcs, thus preserving uniform subdivision despite the non-linear spacing. This approach leverages the of concentric circles to extend the precision of linear transversals to angular measurements, where direct small engravings would be impractical on large-radius arcs. A notable example is Tycho Brahe's implementation on his large mural quadrants at , where two groups of concentric arcs were integrated into the scale: an outer arc for degrees and inner arcs for minutes, connected by transversals to achieve readings down to 5 seconds of arc. This method enhanced the accuracy of his observations, surpassing earlier nonius divisions. Despite these advantages, circular transversals present challenges in fabrication and durability. the intricate secant grid on curved metal surfaces requires high precision, often leading to inconsistencies if not executed by skilled artisans, and the fine lines are susceptible to wear from repeated alidade contact or environmental exposure.

Applications

In Astronomical Instruments

In astronomical instruments of the pre-telescopic era, transversals were employed to achieve high-precision angular measurements, particularly for determining star altitudes and celestial positions. These diagonal scales allowed for subdivisions finer than standard linear divisions, enabling readings down to arcseconds on large instruments. For instance, in quadrants and similar devices, transversals facilitated accurate observations by magnifying scale intervals without requiring optical aids, thus supporting the reform of astronomy through empirical data collection. At Tycho Brahe's observatory, transversals were prominently featured on mural quadrants, such as the Quadrans Maximus with a 5.5-meter radius constructed around 1570, where the brass arc was calibrated using transversal points to resolve angles to 10 arcseconds. These setups, combined with pinhole or pinnule sights, minimized errors by ensuring the observer's eye aligned precisely with the , allowing parallax-free readings during meridian transits of stars. This configuration on the mural quadrant, fixed to a north-south wall, permitted consistent measurements of stellar altitudes essential for cataloging positions. Similarly, Tycho's equatorial armillary spheres, including a 2.6-meter model built in the 1580s, incorporated transversal scales on their rings for fine subdivision, enhancing precision in equatorial coordinate observations while employing sights to further reduce . Earlier examples include circular transversals on astrolabes, as described by Levi ben Gerson in the , which extended the instrument's utility for altitude measurements to sub-degree accuracy. The precision enabled by these transversal-equipped instruments was crucial in the pre-1609 period, yielding observational accuracy of 1-2 arcminutes overall, which provided the robust dataset that used to derive his laws of planetary motion. Tycho's meticulous records, free from significant distortions, marked a pivotal advancement in naked-eye astronomy, surpassing prior limitations and laying the groundwork for heliocentric models.

In Engineering Measurements

Transversals found practical application in and cross-staff instruments, which were employed in and land to measure angles and establish baselines for methods. These tools allowed surveyors to determine distances indirectly through angular observations, facilitating accurate mapping of terrain without direct linear measurement over long spans. For instance, the , developed in the , served as a foundational surveying device for such tasks, enabling portable angle sightings essential for early geodetic work. In engineering drawings, diagonal scales— a form of transversal—were incorporated into rulers and straightedges to convert between units, such as from to millimeters, achieving precisions down to 1/10 . This design magnified subdivisions along an inclined line, permitting finer readings when used with dividers or pointers, which was crucial for scaling plans in and mechanical drafting. As described in 18th-century texts, such scales enabled measurements like "a line of 347, 34 7/10 or 3 47/100" units, supporting detailed technical illustrations without additional instruments. During the , transversals persisted in and protractors for precise length and measurements, offering an alternative to emerging vernier scales in scenarios requiring simplicity and portability. Protractors often featured double diagonal scales for enhanced , while used transversal graduations to gauge small dimensions in workshop settings. These applications highlighted transversals' role in providing high-resolution capabilities through geometric subdivision alone, without the mechanical complexity of later innovations.

References

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