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43-459: Philo of Byzantium ( c. 280 BC – c. 220 BC) encountered and described a weapon similar to the polybolos, a catapult that could fire again and again without a need for manual reloading. Philo left a detailed description of the gears that powered its chain drive (the oldest known application of such a mechanism) and that placed bolt after bolt into its firing slot. The polybolos would have differed from an ordinary ballista in that it had

86-463: A Philo line, and the base of a perpendicular from the apex of the angle to the line, are equidistant from the endpoints of the line. That is, suppose that segment D E {\displaystyle DE} is the Philo line for point P {\displaystyle P} and angle D O E {\displaystyle DOE} , and let Q {\displaystyle Q} be

129-494: A multithrower. In 2010, a MythBusters episode was dedicated to building and testing a replica, and concluded that its existence as a historical weapon was plausible. However, the machine MythBusters built was prone to breakdowns that had to be fixed multiple times. Philo of Byzantium Philo of Byzantium ( Ancient Greek : Φίλων ὁ Βυζάντιος , Phílōn ho Byzántios , c.  280 BC  – c.  220 BC ), also known as Philo Mechanicus ( Latin for "Philo

172-439: A pumice in the process. Once the spoon has emptied, it is pulled up again by the counterweight, closing the door on the pumice by the tightening string. Remarkably, Philo's comment that "its construction is similar to that of clocks" indicates that such escapements mechanism were already integrated in ancient water clocks. He is also credited with the construction of the first thermoscope (or Philo thermometer), an early version of

215-412: A right angle, the limit m → ∞ {\displaystyle m\to \infty } of the previous section results in the following special case: These lines intersect the y {\displaystyle y} -axis at which has the solution The squared Euclidean distance between the intersections of the horizontal line and vertical lines is The Philo Line is defined by

258-401: A wooden hopper magazine, capable of holding several dozen bolts, that was positioned over the mensa (the cradle that holds the bolt prior to firing). The mechanism is unique in that it is driven by a flat-link chain connected to a windlass . The mensa itself was a sliding plank (similar to that on the gastraphetes) containing the claw latches used to pull back the drawstring and was attached to

301-489: Is Given the cubic equation for α {\displaystyle \alpha } above, which is one of the two cubic polynomials in the numerator, this is zero. This is the algebraic proof that the minimization of D E {\displaystyle DE} leads to D Q = P E {\displaystyle DQ=PE} . The equation of a bundle of lines with inclination α {\displaystyle \alpha } that run through

344-476: Is These lines intersect the horizontal axis at which has the solution These lines intersect the opposite side y = m x {\displaystyle y=mx} at which has the solution The squared Euclidean distance between the intersections of the horizontal line and the diagonal is The Philo Line is defined by the minimum of that distance at negative α {\displaystyle \alpha } . An arithmetic expression for

387-464: Is inscribed in the circle with R P {\displaystyle RP} as diameter, it is a right triangle, and V {\displaystyle V} is the base of a perpendicular from the apex of the angle to the Philo line. Let W {\displaystyle W} be the point where line Q R {\displaystyle QR} crosses a perpendicular line through V {\displaystyle V} . Then

430-602: Is printed in R. Hercher 's edition of Aelian (Paris: Firmin Didot, 1858); an English translation by Jean Blackwood is included as an appendix in The Seven Wonders of the World by Michael Ashley (Glasgow: Fontana Paperbacks, 1980). Philo line In geometry , the Philo line is a line segment defined from an angle and a point inside the angle as the shortest line segment through

473-469: Is put into the center of the coordinate system, the direction from O {\displaystyle O} to E {\displaystyle E} defines the horizontal x {\displaystyle x} -coordinate, and the direction from O {\displaystyle O} to D {\displaystyle D} defines the line with the equation y = m x {\displaystyle y{=}mx} in

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516-461: Is similar to the solution given by Hero of Alexandria several centuries later. A treatise titled " Seven Wonders of the World " ( Περὶ τῶν Ἑπτὰ Θεαμάτων , Perì tō̂n Heptà Theamátōn ) is sometimes attributed to this Philo but more probably belongs to a different Philo of Byzantium, distinguished as Philo the Paradoxographer, who lived in a much later date, probably the 4th–5th century AD. It

559-580: Is similarly impossible to construct the Philo line with these tools. Given the point P {\displaystyle P} and the angle D O E {\displaystyle DOE} , a variant of the problem may minimize the area of the triangle O E D {\displaystyle OED} . With the expressions for ( E x , E y ) {\displaystyle (E_{x},E_{y})} and ( D x , D y ) {\displaystyle (D_{x},D_{y})} given above,

602-479: The area is half the product of height and base length, Finding the slope α {\displaystyle \alpha } that minimizes the area means to set ∂ A / ∂ α = 0 {\displaystyle \partial A/\partial \alpha =0} , Again discarding the root α = P y / P x {\displaystyle \alpha =P_{y}/P_{x}} which does not define

645-565: The Engineer"), was a Greek engineer, physicist and writer on mechanics , who lived during the latter half of the 3rd century BC. Although he was from Byzantium he lived most of his life in Alexandria , Egypt. He was probably younger than Ctesibius , though some place him a century earlier. Philo was the author of a large work, the Syntaxis ( Μηχανική Σύνταξη , Mēkhanikḗ Sýntaxē ), which contained

688-460: The Greeks. Philo's works also contain the oldest known application of a chain drive in a repeating crossbow . Two flat-linked chains were connected to a windlass , which by winding back and forth would automatically fire the machine's arrows until its magazine was empty. Philo also was the first to describe a gimbal : an eight-sided ink pot that could be turned any way without spilling and expose

731-472: The Philo line of point P {\displaystyle P} with respect to right angle Q R S {\displaystyle QRS} . Define point V {\displaystyle V} to be the point of intersection of line T U {\displaystyle TU} and of the circle through points P Q R S {\displaystyle PQRS} . Because triangle R V P {\displaystyle RVP}

774-442: The base of a perpendicular line O Q {\displaystyle OQ} to D E {\displaystyle DE} . Then D P = E Q {\displaystyle DP=EQ} and D Q = E P {\displaystyle DQ=EP} . Conversely, if P {\displaystyle P} and Q {\displaystyle Q} are any two points equidistant from

817-413: The chain link. When loading a new bolt and spanning the drawstring, the windlass is rotated counterclockwise by an operator standing on the left side of the weapon; this drives the mensa forward towards the bow string. At the very front, a metal lug triggers the latching claws into catching the drawstring. Once the string is held firm by the trigger mechanism, the windlass is then rotated clockwise; pulling

860-424: The coordinates ( D x , D y ) {\displaystyle (D_{x},D_{y})} shown above, the squared distance from D {\displaystyle D} to Q {\displaystyle Q} is The squared distance from E {\displaystyle E} to P {\displaystyle P} is The difference of these two expressions

903-465: The cube , that is, to construct a geometric representation of the cube root of two, and this was Philo's purpose in defining this line. Specifically, let P Q R S {\displaystyle PQRS} be a rectangle whose aspect ratio P Q : Q R {\displaystyle PQ:QR} is 1 : 2 {\displaystyle 1:2} , as in the figure. Let T U {\displaystyle TU} be

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946-515: The ends of a line segment D E {\displaystyle DE} , and if O {\displaystyle O} is any point on the line through Q {\displaystyle Q} that is perpendicular to D E {\displaystyle DE} , then D E {\displaystyle DE} is the Philo line for angle D O E {\displaystyle DOE} and point P {\displaystyle P} . A suitable fixation of

989-623: The equalities of segments R S = P Q {\displaystyle RS=PQ} , R W = Q U {\displaystyle RW=QU} , and W U = R Q {\displaystyle WU=RQ} follow from the characteristic property of the Philo line. The similarity of the right triangles P Q U {\displaystyle PQU} , R W V {\displaystyle RWV} , and V W U {\displaystyle VWU} follow by perpendicular bisection of right triangles. Combining these equalities and similarities gives

1032-408: The equality of proportions R S : R W = P Q : Q U = R W : W V = W V : W U = W V : R Q {\displaystyle RS:RW=PQ:QU=RW:WV=WV:WU=WV:RQ} or more concisely R S : R W = R W : W V = W V : R Q {\displaystyle RS:RW=RW:WV=WV:RQ} . Since

1075-406: The equation above as α 1 = P y / ( P x − E x ) {\displaystyle \alpha _{1}=P_{y}/(P_{x}-E_{x})} and plugging this into the previous equation one finds that E x {\displaystyle E_{x}} is a root of the cubic polynomial So solving that cubic equation finds

1118-406: The first and last terms of these three equal proportions are in the ratio 1 : 2 {\displaystyle 1:2} , the proportions themselves must all be 1 : 2 3 {\displaystyle 1:{\sqrt[{3}]{2}}} , the proportion that is required to double the cube. Since doubling the cube is impossible with a straightedge and compass construction , it

1161-567: The following sections: The military sections Belopoeica and Poliorcetica are extant in Greek, detailing missiles, the construction of fortresses, provisioning, attack and defence, as are fragments of Isagoge and Automatopoeica (ed. R. Schone, 1893, with German translation in Hermann August Theodor Köchly's Griechische Kriegsschriftsteller , vol. i. 1853; E. A. Rochas d'Aiglun, Poliorcetique des Grecs , 1872). Another portion of

1204-523: The horizontal axis and D = ( D x , D y ) = ( D x , m D x ) {\displaystyle D=(D_{x},D_{y})=(D_{x},mD_{x})} on the other side of the triangle. The equation of a bundle of lines with inclinations α {\displaystyle \alpha } that run through the point ( x , y ) = ( P x , P y ) {\displaystyle (x,y)=(P_{x},P_{y})}

1247-412: The ink on top. This was done by the suspension of the inkwell at the centre, which was mounted on a series of concentric metal rings which remained stationary no matter which way the pot turns. In his Pneumatics (chapter 31) Philo describes an escapement mechanism, the earliest known, as part of a washstand . A counterweighted spoon, supplied by a water tank, tips over in a basin when full releasing

1290-537: The intersection of the Philo line on the horizontal axis. Plugging in the same expression into the expression for the squared distance gives Since the line O Q {\displaystyle OQ} is orthogonal to E D {\displaystyle ED} , its slope is − 1 / α {\displaystyle -1/\alpha } , so the points on that line are y = − x / α {\displaystyle y=-x/\alpha } . The coordinates of

1333-414: The line given the directions from O {\displaystyle O} to E {\displaystyle E} and from O {\displaystyle O} to D {\displaystyle D} and the location of P {\displaystyle P} in that infinite triangle is obtained by the following algebra: The point O {\displaystyle O}

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1376-540: The location of the minimum is obtained by setting the derivative ∂ d 2 / ∂ α = 0 {\displaystyle \partial d^{2}/\partial \alpha =0} , so So calculating the root of the polynomial in the numerator, determines the slope of the particular line in the line bundle which has the shortest length. [The global minimum at inclination α = P y / P x {\displaystyle \alpha =P_{y}/P_{x}} from

1419-412: The mensa back and drawing the bow string with it. At the same time, a round wooden pole in the bottom of the magazine is rotated via a spiral groove being driven by a rivet attached to the sliding mensa; dropping a single bolt from a carved notch in the rotating pole. With the drawstring pulled back and a bolt loaded on the mensa, the polybolos is ready to be fired. As the windlass is rotated further back to

1462-702: The minimum of that curve (at negative α {\displaystyle \alpha } ). An arithmetic expression for the location of the minimum is where the derivative ∂ d 2 / ∂ α = 0 {\displaystyle \partial d^{2}/\partial \alpha =0} , so equivalent to Therefore Alternatively, inverting the previous equations as α 1 = P y / ( P x − E x ) {\displaystyle \alpha _{1}=P_{y}/(P_{x}-E_{x})} and plugging this into another equation above one finds The Philo line can be used to double

1505-420: The point ( x , y ) = ( P x , P y ) {\displaystyle (x,y)=(P_{x},P_{y})} , P x , P y > 0 {\displaystyle P_{x},P_{y}>0} , has an intersection with the x {\displaystyle x} -axis given above. If D O E {\displaystyle DOE} form

1548-558: The point Q = ( Q x , Q y ) {\displaystyle Q=(Q_{x},Q_{y})} are calculated by intersecting this line with the Philo line, y = α ( x − P x ) + P y {\displaystyle y=\alpha (x-P_{x})+P_{y}} . α ( x − P x ) + P y = − x / α {\displaystyle \alpha (x-P_{x})+P_{y}=-x/\alpha } yields With

1591-474: The point that has its endpoints on the two sides of the angle. Also known as the Philon line , it is named after Philo of Byzantium , a Greek writer on mechanical devices, who lived probably during the 1st or 2nd century BC. Philo used the line to double the cube ; because doubling the cube cannot be done by a straightedge and compass construction , neither can constructing the Philo line. The defining point of

1634-560: The rectilinear coordinate system. m {\displaystyle m} is the tangent of the angle in the triangle D O E {\displaystyle DOE} . Then P {\displaystyle P} has the Cartesian Coordinates ( P x , P y ) {\displaystyle (P_{x},P_{y})} and the task is to find E = ( E x , 0 ) {\displaystyle E=(E_{x},0)} on

1677-418: The root of the other factor is not of interest; it does not define a triangle but means that the horizontal line, the diagonal and the line of the bundle all intersect at ( 0 , 0 ) {\displaystyle (0,0)} .] − α {\displaystyle -\alpha } is the tangent of the angle O E D {\displaystyle OED} . Inverting

1720-406: The thermometer. In mathematics, Philo tackled the problem of doubling the cube . The doubling of the cube was necessitated by the following problem: given a catapult, construct a second catapult that is capable of firing a projectile twice as heavy as the projectile of the first catapult. His solution was to find the point of intersection of a rectangular hyperbola and a circle , a solution that

1763-559: The very back end, the claws on the mensa meets another lug like the one that pushed the claws into catching the string. This one causes the claws to disengage the drawstring and automatically fires the loaded bolt. Upon the bolt being fired, the process is repeated. The repetition provides the weapon's name, in Greek πολυβόλος , "throwing many missiles", from πολύς ( polys ), "multiple, many" and -βόλος ( -bolos ) "thrower", in turn from βάλλω ( ballo ), "to throw, to hurl", literally

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1806-480: The work, on pneumatic engines, has been preserved in the form of a Latin translation ( De Ingeniis Spiritualibus ) of an Arabic translation (ed. W. Schmidt, with German translation, in the works of Heron of Alexandria , vol. i., in the Teubner series, 1899; with French translation by Rochas, La Science des philosophes... dans l'antiquité , 1882). Further portions probably survive in a derivative form, incorporated into

1849-420: The works of Vitruvius and of Arabic authors. The Philo line , a geometric construction that can be used to double the cube , is attributed to Philo. According to recent research, a section of Philo's Pneumatics which so far has been regarded as a later Arabic interpolation, includes the first description of a water mill in history, placing the invention of the water mill in the mid-third century BC by

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