97-577: A slide rule is a hand -operated mechanical calculator consisting of slidable rulers for evaluating mathematical operations such as multiplication , division , exponents , roots , logarithms , and trigonometry . It is one of the simplest analog computers . Slide rules exist in a diverse range of styles and generally appear in a linear, circular or cylindrical form. Slide rules manufactured for specialized fields such as aviation or finance typically feature additional scales that aid in specialized calculations particular to those fields. The slide rule
194-427: A lookup table that maps from position on the ruler as each function's input. Calculations that can be reduced to simple addition or subtraction using those precomputed functions can be solved by aligning the two rulers and reading the approximate result. For example, a number to be multiplied on one logarithmic-scale ruler can be aligned with the start of another such ruler to sum their logarithms. Then by applying
291-409: A 10 cm (3.9 in) circular would have a maximum precision approximately equal to a 31.4 cm (12.4 in) ordinary slide rule. Circular slide rules also eliminate "off-scale" calculations, because the scales were designed to "wrap around"; they never have to be reoriented when results are near 1.0—the rule is always on scale. However, for non-cyclical non-spiral scales such as S, T, and LL's,
388-439: A body and a slider that can be slid along within the body and both of these have numerical scales inscribed on them. On duplex rules the body and/or the slider have scales on the back as well as the front. The slider's scales may be visible from the back or the slider may need to be slid right out and replaced facing the other way round. A cursor (also called runner or glass) containing one (or more) hairlines may be slid along
485-438: A calculation are generally done mentally or on paper, not on the slide rule. Most slide rules consist of three parts: Some slide rules ("duplex" models) have scales on both sides of the rule and slide strip, others on one side of the outer strips and both sides of the slide strip (which can usually be pulled out, flipped over and reinserted for convenience), still others on one side only ("simplex" rules). A sliding cursor with
582-405: A compact fist, presumably for fighting purposes. The fist is compact and thus effective as a weapon. It also provides protection for the fingers. However, this is not widely accepted to be one of the primary selective pressures acting on hand morphology throughout human evolution, with tool use and production being thought to be far more influential. Slide rule scale A slide rule scale
679-443: A free dish and one cursor. The dual cursor versions perform multiplication and division by holding a constant angle between the cursors as they are rotated around the dial. The onefold cursor version operates more like the standard slide rule through the appropriate alignment of the scales. The basic advantage of a circular slide rule is that the widest dimension of the tool was reduced by a factor of about 3 (i.e. by π ). For example,
776-399: A higher degree, the hands of other primates are anatomically similar and the dexterity of the human hand can not be explained solely on anatomical factors. The neural machinery underlying hand movements is a major contributing factor; primates have evolved direct connections between neurons in cortical motor areas and spinal motoneurons , giving the cerebral cortex monosynaptic control over
873-523: A pencil—reflects individual brain functioning. Among humans, the hands play an important function in body language and sign language . Likewise, the ten digits of two hands and the twelve phalanges of four fingers (touchable by the thumb) have given rise to number systems and calculation techniques. Many mammals and other animals have grasping appendages similar in form to a hand such as paws , claws , and talons, but these are not scientifically considered to be grasping hands. The scientific use of
970-399: A primitive trait, while the palms of other extant higher primates are elongated to the extent that some of the thumb's original function has been lost (most notably in highly arboreal primates such as the spider monkey ). In humans, the big toe is thus more derived than the thumb. There is a hypothesis suggesting the form of the modern human hand is especially conducive to the formation of
1067-427: A transitional element to the fifth metacarpal. Together with the thumb, the four fingers form four oblique arches, of which the arch of the index finger functionally is the most important, especially for precision grip, while the arch of the little finger contribute an important locking mechanism for power grip. The thumb is undoubtedly the "master digit" of the hand, giving value to all the other fingers. Together with
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#17327934192251164-415: A vertical alignment line is used to find corresponding points on scales that are not adjacent to each other or, in duplex models, are on the other side of the rule. The cursor can also record an intermediate result on any of the scales. Scales may be grouped in decades , where each decade corresponds to a range of numbers that spans a ratio of 10 (i.e. a range from 10 to 10). For example, the range 1 to 10
1261-441: A wooden body within which a slider could be fitted and adjusted. Simple slide rules will have a C and D scale for multiplication and division , most likely an A and B for squares and square roots , and possibly CI and K for reciprocals and cubes . In the early days of slide rules few scales were provided and no labelling was necessary. However, gradually the number of scales tended to increase. Amédée Mannheim introduced
1358-489: A workaround is possible. The rule illustrated is an Aristo 0972 HyperLog, which has 31 scales. The scales in the table below are those appropriate for general mathematical use rather than for specific professions. Gauge marks are often added to the scales either marking important constants (e.g. π at 3.14159) or useful conversion coefficients (e.g. ρ " at 180*60*60/π or 206.3x10 to find sine and tan of small angles ). A cursor may have subsidiary hairlines beside
1455-413: Is a prehensile , multi- fingered appendage located at the end of the forearm or forelimb of primates such as humans , chimpanzees , monkeys , and lemurs . A few other vertebrates such as the koala (which has two opposable thumbs on each "hand" and fingerprints extremely similar to human fingerprints ) are often described as having "hands" instead of paws on their front limbs. The raccoon
1552-439: Is a line with graduated markings inscribed along the length of a slide rule used for mathematical calculations. The earliest such device had a single logarithmic scale for performing multiplication and division, but soon an improved technique was developed which involved two such scales sliding alongside each other. Later, multiple scales were provided with the most basic being logarithmic but with others graduated according to
1649-870: Is a single decade, and the range from 10 to 100 is another decade. Thus, single-decade scales (named C and D) range from 1 to 10 across the entire length of the slide rule, while double-decade scales (named A and B) range from 1 to 100 over the length of the slide rule. The following logarithmic identities transform the operations of multiplication and division to addition and subtraction, respectively: log ( x × y ) = log ( x ) + log ( y ) , {\displaystyle \log(x\times y)=\log(x)+\log(y)\,,} log ( x / y ) = log ( x ) − log ( y ) . {\displaystyle \log(x/y)=\log(x)-\log(y)\,.} With two logarithmic scales,
1746-425: Is again positioned to start at the 2 on the bottom scale. Since 2 represents 20 , all numbers in that scale are multiplied by 10 . Thus, any answer in the second set of numbers is multiplied by 100 . Since 8.8 in the top scale represents 88 , the answer must additionally be multiplied by 10 . The answer directly reads 1.76 . Multiply by 100 and then by 10 to get the actual answer: 1,760 . In general,
1843-464: Is closely related to nomograms used for application-specific computations. Though similar in name and appearance to a standard ruler , the slide rule is not meant to be used for measuring length or drawing straight lines. Nor is it designed for addition or subtraction, which is usually performed using other methods, like using an abacus . Maximum accuracy for standard linear slide rules is about three decimal significant digits, while scientific notation
1940-503: Is for base e. Logarithms to any other base can be calculated by reversing the procedure for calculating powers of a number. For example, log2 values can be determined by lining up either leftmost or rightmost 1 on the C scale with 2 on the LL2 scale, finding the number whose logarithm is to be calculated on the corresponding LL scale, and reading the log2 value on the C scale. Addition and subtraction aren't typically performed on slide rules, but
2037-474: Is located on one of the sides, parallel to the arm. A reliable way of identifying human hands is from the presence of opposable thumbs. Opposable thumbs are identified by the ability to be brought opposite to the fingers, a muscle action known as opposition. The skeleton of the human hand consists of 27 bones: the eight short carpal bones of the wrist are organized into a proximal row ( scaphoid , lunate , triquetral and pisiform ) which articulates with
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#17327934192252134-610: Is possible using either of the following two techniques: Using (almost) any strictly monotonic scales , other calculations can also be made with one movement. For example, reciprocal scales can be used for the equality 1 x + 1 y = 1 z {\displaystyle {\frac {1}{x}}+{\frac {1}{y}}={\frac {1}{z}}} (calculating parallel resistances , harmonic mean , etc.), and quadratic scales can be used to solve x 2 + y 2 = z 2 {\displaystyle x^{2}+y^{2}=z^{2}} . The width of
2231-408: Is supplemented by the precision grip between the thumb and the distal finger pads made possible by the opposable thumbs. Hominidae (great apes including humans) acquired an erect bipedal posture about 3.6 million years ago , which freed the hands from the task of locomotion and paved the way for the precision and range of motion in human hands. Functional analyses of the features unique to
2328-497: Is to the neck of a metacarpal. One can also have a broken finger . The prehensile hands and feet of primates evolved from the mobile hands of semi- arboreal tree shrews that lived about 60 million years ago . This development has been accompanied by important changes in the brain and the relocation of the eyes to the front of the face, together allowing the muscle control and stereoscopic vision necessary for controlled grasping. This grasping, also known as power grip,
2425-561: Is used to keep track of the order of magnitude of results. English mathematician and clergyman Reverend William Oughtred and others developed the slide rule in the 17th century based on the emerging work on logarithms by John Napier . It made calculations faster and less error-prone than evaluating on paper . Before the advent of the scientific pocket calculator , it was the most commonly used calculation tool in science and engineering . The slide rule's ease of use, ready availability, and low cost caused its use to continue to grow through
2522-505: Is usually described as having "hands" though opposable thumbs are lacking. Some evolutionary anatomists use the term hand to refer to the appendage of digits on the forelimb more generally—for example, in the context of whether the three digits of the bird hand involved the same homologous loss of two digits as in the dinosaur hand. The human hand usually has five digits: four fingers plus one thumb ; these are often referred to collectively as five fingers , however, whereby
2619-452: The 1 on the top is moved to a factor on the bottom, and the answer is read off the bottom where the other factor is on the top. This works because the distances from the 1 mark are proportional to the logarithms of the marked values. The illustration below demonstrates the computation of 5.5 / 2 . The 2 on the top scale is placed over the 5.5 on the bottom scale. The resulting quotient, 2.75 , can then be read below
2716-473: The carpal tunnel and contribute to the deep and superficial palmar arches . Several muscle tendons attaching to the TCL and the distal carpals also contribute to maintaining the carpal arch. Compared to the carpal arches, the arch formed by the distal ends of the metacarpal bones is flexible due to the mobility of the peripheral metacarpals (thumb and little finger). As these two metacarpals approach each other,
2813-600: The chimpanzee–human last common ancestor (CHLCA) and absent in modern humans are still present in the hands of Australopithecus , Paranthropus , and Homo floresiensis . This suggests that the derived changes in modern humans and Neanderthals did not evolve until 2.5 to 1.5 million years ago or after the appearance of the earliest Acheulian stone tools, and that these changes are associated with tool-related tasks beyond those observed in other hominins. The thumbs of Ardipithecus ramidus , an early hominin, are almost as robust as in humans, so this may be
2910-419: The dermis of palmoplantar skin inhibit melanin production and thus the ability to tan , and promote the thickening of the stratum lucidum and stratum corneum layers of the epidermis . All parts of the skin involved in grasping are covered by papillary ridges ( fingerprints ) acting as friction pads. In contrast, the hairy skin on the dorsal side is thin, soft, and pliable, so that the skin can recoil when
3007-481: The law of the logarithm of a product , the product of the two numbers can be read. More elaborate slide rules can perform other calculations, such as square roots , exponentials , logarithms , and trigonometric functions . The user may estimate the location of the decimal point in the result by mentally interpolating between labeled graduations. Scientific notation is used to track the decimal point for more precise calculations. Addition and subtraction steps in
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3104-523: The ulnar nerve may result in a condition in which some of the fingers cannot be flexed. A common fracture of the hand is a scaphoid fracture —a fracture of the scaphoid bone , one of the carpal bones. This is the commonest carpal bone fracture and can be slow to heal due to a limited blood flow to the bone. There are various types of fracture to the base of the thumb; these are known as Rolando fractures , Bennet's fracture , and Gamekeeper's thumb . Another common fracture, known as Boxer's fracture ,
3201-478: The 1930s for aircraft pilots to help with dead reckoning . With the aid of scales printed on the frame it also helps with such miscellaneous tasks as converting time, distance, speed, and temperature values, compass errors, and calculating fuel use. The so-called "prayer wheel" is still available in flight shops, and remains widely used. While GPS has reduced the use of dead reckoning for aerial navigation, and handheld calculators have taken over many of its functions,
3298-543: The 1950s and 1960s, even as desktop electronic computers were gradually introduced. But after the handheld scientific calculator was introduced in 1972 and became inexpensive in the mid-1970s, slide rules became largely obsolete , so most suppliers departed the business. In the United States , the slide rule is colloquially called a slipstick . Each ruler's scale has graduations labeled with precomputed outputs of various mathematical functions , acting as
3395-406: The 2 of the bottom scale, and then reading the marking 1.4 off the bottom two-decade scale where 7 is on the top scale: [REDACTED] But since the 7 is above the second set of numbers that number must be multiplied by 10 . Thus, even though the answer directly reads 1.4 , the correct answer is 1.4×10 = 14 . For an example with even larger numbers, to multiply 88×20 , the top scale
3492-419: The A scale). Slide the slide until the number on the D scale which is against 1 on the C cursor is the same as the number on the B cursor which is against the base number on the A scale. (Examples: A 8, B 2, C 1, D 2; A 27, B 3, C 1, D 3.) Quadratic equations of the form a x 2 + b x + c = 0 {\displaystyle ax^{2}+bx+c=0} can be solved by first reducing
3589-409: The A, B, C and D labels in 1859 and, after that, manufacturers began to adopt a somewhat standardised, though idiosyncratic, system of labels so the various scales could be quickly identified. Advanced slide rules have many scales and they are often designed with particular types of user in mind, for example electrical engineers or surveyors. There are rarely scales for addition and subtraction but
3686-585: The CI scale is used. Common forms such as k sin x {\displaystyle k\sin x} can be read directly from x on the S scale to the result on the D scale, when the C scale index is set at k . For angles below 5.7 degrees, sines, tangents, and radians are approximately equal, and are found on the ST or SRT (sines, radians, and tangents) scale, or simply divided by 57.3 degrees/ radian . Inverse trigonometric functions are found by reversing
3783-466: The D scale and read its square on the A scale. Inverting this process allows square roots to be found, and similarly for the powers 3, 1/3, 2/3, and 3/2. Care must be taken when the base, x, is found in more than one place on its scale. For instance, there are two nines on the A scale; to find the square root of nine, use the first one; the second one gives the square root of 90. For x y {\displaystyle x^{y}} problems, use
3880-411: The D scale. The cursor is then moved along the rule until a position is found where the numbers on the CI and D scales add up to p {\displaystyle p} . These two values are the roots of the equation. The LLN scales can be used to compute and compare the cost or return on a fixed rate loan or investment. The simplest case is for continuously compounded interest. Example: Taking D as
3977-481: The E6B remains widely used as a primary or backup device and the majority of flight schools demand that their students have some degree of proficiency in its use. Proportion wheels are simple circular slide rules used in graphic design to calculate aspect ratios . Lining up the original and desired size values on the inner and outer wheels will display their ratio as a percentage in a small window. Though not as common since
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4074-495: The LL scales. When several LL scales are present, use the one with x on it. First, align the leftmost 1 on the C scale with x on the LL scale. Then, find y on the C scale and go down to the LL scale with x on it. That scale will indicate the answer. If y is "off the scale", locate x y / 2 {\displaystyle x^{y/2}} and square it using the A and B scales as described above. Alternatively, use
4171-480: The S scale with C (or D) scale. (On many closed-body rules the S scale relates to the A and B scales instead and covers angles from around 0.57 up to 90 degrees; what follows must be adjusted appropriately.) The S scale has a second set of angles (sometimes in a different color), which run in the opposite direction, and are used for cosines. Tangents are found by comparing the T scale with the C (or D) scale for angles less than 45 degrees. For angles greater than 45 degrees
4268-452: The act of positioning the top scale to start at the bottom scale's label for x {\displaystyle x} corresponds to shifting the top logarithmic scale by a distance of log ( x ) {\displaystyle \log(x)} . This aligns each top scale's number y {\displaystyle y} at offset log ( y ) {\displaystyle \log(y)} with
4365-618: The advent of computerized layout, they are still made and used. In 1952, Swiss watch company Breitling introduced a pilot's wristwatch with an integrated circular slide rule specialized for flight calculations: the Breitling Navitimer . The Navitimer circular rule, referred to by Breitling as a "navigation computer", featured airspeed , rate /time of climb/descent, flight time, distance, and fuel consumption functions, as well as kilometer— nautical mile and gallon—liter fuel amount conversion functions. Hand A hand
4462-402: The back of the hand), the deep palmar arch , and the superficial palmar arch . Together these three arches and their anastomoses provide oxygenated blood to the palm, the fingers, and the thumb. The hand is drained by the dorsal venous network of the hand with deoxygenated blood leaving the hand via the cephalic vein and the basilic vein . The glabrous (hairless) skin on the front of
4559-418: The bones of the forearm, and a distal row ( trapezium , trapezoid , capitate and hamate ), which articulates with the bases of the five metacarpal bones of the hand. The heads of the metacarpals will each in turn articulate with the bases of the proximal phalanx of the fingers and thumb. These articulations with the fingers are the metacarpophalangeal joints known as the knuckles. At the palmar aspect of
4656-564: The bottom scale's number at position log ( x ) + log ( y ) {\displaystyle \log(x)+\log(y)} . Because log ( x ) + log ( y ) = log ( x × y ) {\displaystyle \log(x)+\log(y)=\log(x\times y)} , the mark on the bottom scale at that position corresponds to x × y {\displaystyle x\times y} . With x=2 and y=3 for example, by positioning
4753-437: The densest areas of nerve endings in the body, and are the richest source of tactile feedback. They also have the greatest positioning capability of the body; thus, the sense of touch is intimately associated with hands. Like other paired organs (eyes, feet, legs) each hand is dominantly controlled by the opposing brain hemisphere , so that handedness —the preferred hand choice for single-handed activities such as writing with
4850-467: The difficulty in locating figures along a dish, and limited number of scales. Another drawback of circular slide rules is that less-important scales are closer to the center, and have lower precisions. Most students learned slide rule use on the linear slide rules, and did not find reason to switch. One slide rule remaining in daily use around the world is the E6B . This is a circular slide rule first created in
4947-519: The disorders that can cause this is Catel-Manzke syndrome . The fingers may be fused in a disorder known as syndactyly . Or there may be an absence of one or more central fingers—a condition known as ectrodactyly . Additionally, some people are born without one or both hands ( amelia ). Hereditary multiple exostoses of the forearm—also known as hereditary multiple osteochondromas—is another cause of hand and forearm deformity in children and adults. There are several cutaneous conditions that can affect
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#17327934192255044-474: The divisions mark a scale to a precision of two significant figures , and the user estimates the third figure. Some high-end slide rules have magnifier cursors that make the markings easier to see. Such cursors can effectively double the accuracy of readings, permitting a 10-inch slide rule to serve as well as a 20-inch model. Various other conveniences have been developed. Trigonometric scales are sometimes dual-labeled, in black and red, with complementary angles,
5141-423: The equation to the form x 2 − p x + q = 0 {\displaystyle x^{2}-px+q=0} (where p = − b / a {\displaystyle p=-b/a} and q = c / a {\displaystyle q=c/a} ), and then aligning the index ("1") of the C scale to the value q {\displaystyle q} on
5238-447: The exact number varies between people: whereas a pair of sesamoid bones are found at virtually all thumb metacarpophalangeal joints, sesamoid bones are also common at the interphalangeal joint of the thumb (72.9%) and at the metacarpophalangeal joints of the little finger (82.5%) and the index finger (48%). In rare cases, sesamoid bones have been found in all the metacarpophalangeal joints and all distal interphalangeal joints except that of
5335-426: The fingers and toes". The ratio of the length of the index finger to the length of the ring finger in adults is affected by the level of exposure to male sex hormones of the embryo in utero . This digit ratio is below 1 for both sexes but it is lower in males than in females on average. A number of genetic disorders affect the hand. Polydactyly is the presence of more than the usual number of fingers. One of
5432-434: The fingers are stretched. On the dorsal side, the skin can be moved across the hand up to 3 cm (1.2 in); an important input the cutaneous mechanoreceptors . The web of the hand is a "fold of skin which connects the digits". These webs, located between each set of digits, are known as skin folds (interdigital folds or plica interdigitalis). They are defined as "one of the folds of skin, or rudimentary web, between
5529-417: The first metacarpophalangeal joints are small, almost spherical bones called the sesamoid bones. The fourteen phalanges make up the fingers and thumb, and are numbered I-V (thumb to little finger) when the hand is viewed from an anatomical position (palm up). The four fingers each consist of three phalanx bones: proximal, middle, and distal. The thumb only consists of a proximal and distal phalanx. Together with
5626-418: The groove on the dorsum of inferior side of ulna. The hand is innervated by the radial , median , and ulnar nerves . The radial nerve supplies the finger extensors and the thumb abductor , thus the muscles that extends at the wrist and metacarpophalangeal joints (knuckles); and that abducts and extends the thumb. The median nerve supplies the flexors of the wrist and digits, the abductors and opponens of
5723-456: The hand including the nails . The autoimmune disease rheumatoid arthritis can affect the hand, particularly the joints of the fingers. Some conditions can be treated by hand surgery . These include carpal tunnel syndrome , a painful condition of the hand and fingers caused by compression of the median nerve , and Dupuytren's contracture , a condition in which fingers bend towards the palm and cannot be straightened. Similarly, injury to
5820-457: The hand of modern humans have shown that they are consistent with the stresses and requirements associated with the effective use of paleolithic stone tools. It is possible that the refinement of the bipedal posture in the earliest hominids evolved to facilitate the use of the trunk as leverage in accelerating the hand. While the human hand has unique anatomical features, including a longer thumb and fingers that can be controlled individually to
5917-448: The hand, that of the thumb is the most mobile (and the least longitudinal). While the ray formed by the little finger and its associated metacarpal bone still offers some mobility, the remaining rays are firmly rigid. The phalangeal joints of the index finger, however, offer some independence to its finger, due to the arrangement of its flexor and extension tendons. The carpal bones form two transversal rows, each forming an arch concave on
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#17327934192256014-400: The hand, the palm, is relatively thick and can be bent along the hand's flexure lines where the skin is tightly bound to the underlying tissue and bones. Compared to the rest of the body's skin, the hands' palms (as well as the soles of the feet ) are usually lighter—and even much lighter in dark-skinned individuals, compared to the other side of the hand. Indeed, genes specifically expressed in
6111-461: The human hand are plesiomorphic (shared by both ancestors and extant primate species); the elongated thumbs and short hands more closely resemble the hand proportions of Miocene apes than those of extant primates. Humans did not evolve from knuckle-walking apes, and chimpanzees and gorillas independently acquired elongated metacarpals as part of their adaptation to their modes of locomotion. Several primitive hand features most likely present in
6208-413: The human hand include: There are five digits attached to the hand, notably with a nail fixed to the end in place of the normal claw . The four fingers can be folded over the palm which allows the grasping of objects. Each finger, starting with the one closest to the thumb, has a colloquial name to distinguish it from the others: The thumb (connected to the first metacarpal bone and trapezium )
6305-413: The human hand, including pentadactyly (having five fingers), the hairless skin of the palm and fingers, and the os centrale found in human embryos, prosimians, and apes. Furthermore, the precursors of the intrinsic muscles of the hand are present in the earliest fishes, reflecting that the hand evolved from the pectoral fin and thus is much older than the arm in evolutionary terms. The proportions of
6402-450: The index and middle finger, it forms the dynamic tridactyl configuration responsible for most grips not requiring force. The ring and little fingers are more static, a reserve ready to interact with the palm when great force is needed. The muscles acting on the hand can be subdivided into two groups: the extrinsic and intrinsic muscle groups. The extrinsic muscle groups are the long flexors and extensors . They are called extrinsic because
6499-421: The index finger. For example, in some individuals, the ulnar nerve supplies the entire ring finger and the ulnar side of the middle finger, whilst, in others, the median nerve supplies the entire ring finger. The hand is supplied with blood from two arteries, the ulnar artery and the radial artery . These arteries form three arches over the dorsal and palmar aspects of the hand, the dorsal carpal arch (across
6596-520: The interest rate in percent, slide the index (the "1" at the right or left end of the scale) of C to the percent on D. The corresponding value on LL2 directly below the index will be the multiplier for 10 cycles of interest (typically years). The value on LL2 below 2 on the C scale will be the multiplier after 20 cycles, and so on. The S, T, and ST scales are used for trig functions and multiples of trig functions, for angles in degrees. For angles from around 5.7 up to 90 degrees, sines are found by comparing
6693-480: The interlocking shapes of the carpal bones, and the wrist is therefore more stable in flexion than in extension. The distal carpal arch affects the function of the CMC joints and the hands, but not the function of the wrist or the proximal carpal arch. The ligaments that maintain the distal carpal arches are the transverse carpal ligament and the intercarpal ligaments (also oriented transversally). These ligaments also form
6790-644: The log of a value on a multiplier scale), natural logarithm (ln) and exponential ( e ) scales. Others feature scales for calculating hyperbolic functions . On linear rules, the scales and their labeling are highly standardized, with variation usually occurring only in terms of which scales are included and in what order. The Binary Slide Rule manufactured by Gilson in 1931 performed an addition and subtraction function limited to fractions. There are single-decade (C and D), double-decade (A and B), and triple-decade (K) scales. To compute x 2 {\displaystyle x^{2}} , for example, locate x on
6887-420: The long finger. The articulations are: The fixed and mobile parts of the hand adapt to various everyday tasks by forming bony arches: longitudinal arches (the rays formed by the finger bones and their associated metacarpal bones), transverse arches (formed by the carpal bones and distal ends of the metacarpal bones), and oblique arches (between the thumb and four fingers): Of the longitudinal arches or rays of
6984-474: The mathematical function required. Few slide rules have been designed for addition and subtraction, rather the main scales are used for multiplication and division and the other scales are for mathematical calculations involving trigonometric , exponential and, generally, transcendental functions . Before they were superseded by electronic calculators in the 1970s, slide rules were an important type of portable calculating instrument. A slide rule consists of
7081-456: The motoneurons of the hand muscles; placing the hands "closer" to the brain. The recent evolution of the human hand is thus a direct result of the development of the central nervous system , and the hand, therefore, is a direct tool of our consciousness—the main source of differentiated tactile sensations—and a precise working organ enabling gestures—the expressions of our personalities. There are nevertheless several primitive features left in
7178-455: The muscle belly is located on the forearm. The intrinsic muscle groups are the thenar (thumb) and hypothenar (little finger) muscles; the interosseous muscles ( four dorsally and three volarly ) originating between the metacarpal bones; and the lumbrical muscles arising from the deep flexor (and are special because they have no bony origin) to insert on the dorsal extensor hood mechanism. The fingers have two long flexors, located on
7275-593: The other scale, the user can see that at the same time 1.5 is over 2, 2.25 is over 3, 3 is over 4, 3.75 is over 6, 4.5 is over 6, and 6 is over 8, among other pairs. For a real-life situation where 750 represents a whole 100%, these readings could be interpreted to suggest that 150 is 20%, 225 is 30%, 300 is 40%, 375 is 50%, 450 is 60%, and 600 is 80%. In addition to the logarithmic scales, some slide rules have other mathematical functions encoded on other auxiliary scales. The most popular are trigonometric , usually sine and tangent , common logarithm (log 10 ) (for taking
7372-458: The outer rings. Rather than "split" scales, high-end circular rules use spiral scales for more complex operations like log-of-log scales. One eight-inch premium circular rule had a 50-inch spiral log-log scale. Around 1970, an inexpensive model from B. C. Boykin (Model 510) featured 20 scales, including 50-inch C-D (multiplication) and log scales. The RotaRule featured a friction brake for the cursor. The main disadvantages of circular slide rules are
7469-403: The palmar gutter deepens. The central-most metacarpal (middle finger) is the most rigid. It and its two neighbors are tied to the carpus by the interlocking shapes of the metacarpal bones. The thumb metacarpal only articulates with the trapezium and is therefore completely independent, while the fifth metacarpal (little finger) is semi-independent with the fourth metacarpal (ring finger) which forms
7566-407: The palmar side of the thumb, index, middle, and half ring fingers. Dorsal branches innervates the distal phalanges of the index, middle, and half ring fingers. The ulnar nerve supplies the ulnar third of the hand, both at the palm and the back of the hand, and the little and half ring fingers. There is a considerable variation to this general pattern, except for the little finger and volar surface of
7663-410: The palmar side. Because the proximal arch simultaneously has to adapt to the articular surface of the radius and to the distal carpal row, it is by necessity flexible. In contrast, the capitate, the "keystone" of the distal arch, moves together with the metacarpal bones and the distal arch is therefore rigid. The stability of these arches is more dependent of the ligaments and capsules of the wrist than of
7760-415: The phalanges of the fingers and thumb these metacarpal bones form five rays or poly-articulated chains. Because supination and pronation (rotation about the axis of the forearm) are added to the two axes of movements of the wrist, the ulna and radius are sometimes considered part of the skeleton of the hand. There are numerous sesamoid bones in the hand, small ossified nodes embedded in tendons;
7857-404: The process. Many slide rules have S, T, and ST scales marked with degrees and minutes (e.g. some Keuffel and Esser models (Doric duplex 5" models, for example), late-model Teledyne-Post Mannheim-type rules). So-called decitrig models use decimal fractions of degrees instead. Base-10 logarithms and exponentials are found using the L scale, which is linear. Some slide rules have a Ln scale, which
7954-456: The result, 8.25 , can be read beneath the 3 in the top scale in the figure above, without the need to register the intermediate result for 5.5 / 2 . Because pairs of numbers that are aligned on the logarithmic scales form constant ratios, no matter how the scales are offset, slide rules can be used to generate equivalent fractions that solve proportion and percent problems. For example, setting 7.5 on one scale over 10 on
8051-469: The rightmost 1 on the C scale, and read the answer off the next higher LL scale. For example, aligning the rightmost 1 on the C scale with 2 on the LL2 scale, 3 on the C scale lines up with 8 on the LL3 scale. To extract a cube root using a slide rule with only C/D and A/B scales, align 1 on the B cursor with the base number on the A scale (taking care as always to distinguish between the lower and upper halves of
8148-452: The scale width is narrowed to make room for end margins. Circular slide rules are mechanically more rugged and smoother-moving, but their scale alignment precision is sensitive to the centering of a central pivot; a minute 0.1 mm (0.0039 in) off-centre of the pivot can result in a 0.2 mm (0.0079 in) worst case alignment error. The pivot does prevent scratching of the face and cursors. The highest accuracy scales are placed on
8245-439: The slide rule is quoted in terms of the nominal width of the scales. Scales on the most common "10-inch" models are actually 25 cm, as they were made to metric standards, though some rules offer slightly extended scales to simplify manipulation when a result overflows. Pocket rules are typically 5 inches (12 cm). Models a couple of metres (yards) wide were made to be hung in classrooms for teaching purposes. Typically
8342-582: The so-called "Darmstadt" style. Duplex slide rules often duplicate some of the scales on the back. Scales are often "split" to get higher accuracy. For example, instead of reading from an A scale to a D scale to find a square root, it may be possible to read from a D scale to an R1 scale running from 1 to square root of 10 or to an R2 scale running from square root of 10 to 10, where having more subdivisions marked can result in being able to read an answer with one more significant digit. Circular slide rules come in two basic types, one with two cursors, and another with
8439-403: The tendons of these form the anatomical snuff box . Also, the index finger and the little finger have an extra extensor used, for instance, for pointing. The extensors are situated within 6 separate compartments. The first four compartments are located in the grooves present on the dorsum of inferior side of radius while the 5th compartment is in between radius and ulna. The 6th compartment is in
8536-410: The term hand in this sense to distinguish the terminations of the front paws from the hind ones is an example of anthropomorphism . The only true grasping hands appear in the mammalian order of primates . Hands must also have opposable thumbs , as described later in the text. The hand is located at the distal end of each arm. Apes and monkeys are sometimes described as having four hands, because
8633-405: The thumb in opposition, making grasping possible. The extensors are located on the back of the forearm and are connected in a more complex way than the flexors to the dorsum of the fingers. The tendons unite with the interosseous and lumbrical muscles to form the extensorhood mechanism. The primary function of the extensors is to straighten out the digits. The thumb has two extensors in the forearm;
8730-415: The thumb is included as one of the fingers . It has 27 bones, not including the sesamoid bone , the number of which varies among people, 14 of which are the phalanges ( proximal , intermediate and distal ) of the fingers and thumb. The metacarpal bones connect the fingers and the carpal bones of the wrist . Each human hand has five metacarpals and eight carpal bones. Fingers contain some of
8827-461: The thumb, the first and second lumbrical. The ulnar nerve supplies the remaining intrinsic muscles of the hand. All muscles of the hand are innervated by the brachial plexus (C5–T1) and can be classified by innervation: The radial nerve supplies the skin on the back of the hand from the thumb to the ring finger and the dorsal aspects of the index, middle, and half ring fingers as far as the proximal interphalangeal joints. The median nerve supplies
8924-410: The toes are long and the hallux is opposable and looks more like a thumb , thus enabling the feet to be used as hands. The word "hand" is sometimes used by evolutionary anatomists to refer to the appendage of digits on the forelimb such as when researching the homology between the three digits of the bird hand and the dinosaur hand. An adult human male's hand weighs about a pound. Areas of
9021-405: The top scale to start at the bottom scale's 2 , the result of the multiplication 3×2=6 can then be read on the bottom scale under the top scale's 3 : [REDACTED] While the above example lies within one decade, users must mentally account for additional zeroes when dealing with multiple decades. For example, the answer to 7×2=14 is found by first positioning the top scale to start above
9118-534: The top scale's 1 : [REDACTED] There is more than one method for doing division, and the method presented here has the advantage that the final result cannot be off-scale, because one has a choice of using the 1 at either end. With more complex calculations involving multiple factors in the numerator and denominator of an expression, movement of the scales can be minimized by alternating divisions and multiplications. Thus 5.5×3 / 2 would be computed as 5.5 / 2 ×3 and
9215-435: The underside of the forearm. They insert by tendons to the phalanges of the fingers. The deep flexor attaches to the distal phalanx, and the superficial flexor attaches to the middle phalanx. The flexors allow for the actual bending of the fingers. The thumb has one long flexor and a short flexor in the thenar muscle group. The human thumb also has other muscles in the thenar group ( opponens and abductor brevis muscle ), moving
9312-464: The whole rule so that corresponding readings, front and back, can be taken from the various scales on the body and slider. In about 1620, Edmund Gunter introduced what is now known as Gunter's line as one element of the Gunter's sector he invented for mariners. The line, inscribed on wood, was a single logarithmic scale going from 1 to 100. It had no sliding parts but by using a pair of dividers it
9409-403: Was possible to multiply and divide numbers. The form with a single logarithmic scale eventually developed into such instruments as Fuller's cylindrical slide rule . In about 1622, but not published until 1632, William Oughtred invented linear and circular slide rules which had two logarithmic scales that slid beside each other to perform calculations. In 1654 the linear design was developed into
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