Parabola

Parabola (pronounced puh-rab-uh-luh)

(1) In geometry, a plane curve formed by the intersection of a right circular cone with a plane parallel to a generator of the cone; the set of points in a plane that are equidistant from a fixed line and a fixed point in the same plane or in a parallel plane. Equation: y2 = 2px or x2 = 2py.

(2) In rhetoric, the explicit drawing of a parallel between two essentially dissimilar things, especially with a moral or didactic purpose; a parable.

1570s: From the Modern Latin parabola, from the Late Greek παραβολή (parabolḗ) (a comparison; a setting alongside; parable (literally "a throwing beside" hence "a juxtaposition") so called by Apollonius of Perga circa 210 BC because it is produced by "application" of a given area to a given straight line.  The Greek parabolḗ was derived from παραβάλλω (parabállō) (I set side by side”), from παρά (pará) (beside) + βάλλω (bállō) (I throw); a doublet of parable, parole, and palaver.  It had a different sense in Pythagorean geometry.  The adjectival form parabolic (figurative, allegorical, of or pertaining to a parable) from the Medieval Latin parabolicus from the Late Greek parabolikos (figurative) from parabolē (comparison) is now probably the most widely used.  In geometry, in the sense of “pertaining to a parabola”, it’s been in use since 1702.  A parabola is a curve formed by the set of points in a plane that are all equally distant from both a given line (called the directrix) and a given point (called the focus) that is not on the line.  It’s best visualised as a shape consisting of a single bend and two lines going off to an infinite distance.

Monza

On the Monza banking: Maserati 250F (left), Ferrari F555 Supersqualo (centre) & Vanwall VW2 (right).

The Autodromo Nazionale di Monza (National Automobile Racetrack of Monza) is now the fastest circuit still used in Formula One, the highest recorded speed the 231.5 mph (372.6 km/h) attained during qualifying for the 2005 Italian Grand Prix by a McLaren-Mercedes MP4-20 (in qualifying trim) on the long straight between the Lesmo corners and the Variante del Rettifilo.  Built in 1922, the Italian Grand Prix has been held there every year since 1949 except in 1980 when the track was being modernised and it’s a wonder the track has survived the attention of the Fédération Internationale de l'Automobile (the FIA; the International Automobile Federation).  Once an admirable body, the FIA has in recent decades degenerated into international sport’s dopiest regulatory body and has for some yers attempted to make motorsport as slow, quiet and processional as possible, issues like DEI (diversity, equity and inclusion) now apparently more important than quality of racing.  Set in the Royal Villa of Monza park and surrounded by forest, the complex is configured as three tracks: the 3.6 mile (5.8 kilometre) Grand Prix track, the 1.5 mile (2.4 kilometre) short circuit and the 2.6 mile (4.3 kilometre) high speed oval track with its famous steep bankings which was unused for decades left to fall into disrepair before it was restored in the 2010s.  The major features of the main Grand Prix track include the Curva Grande, the Curva di Lesmo, the Variante Ascari and the famous Curva Parabolica.

On the parabolica: 1966 Italian Grand Prix.

The Curva Parabolica (universally known as “the parabolica”) is the circuit’s signature corner, an increasing radius, long right-hand turn and the final corner before the main straight so the speed one can attain on the straight is determined essentially by the exit speed from the the parabolica; a perfect execution is thus essential for a quick lap.  Although in motorsport it’s common to discuss the lengths of straights, one notable statistic is that even at close to 150 mph (200 km/h) speed with with the fastest cars take the curve, to transit the the parabolica takes just over 7.6 seconds.  Improvements to both the cars and the circuit means it’s now a less dangerous place but many drivers have died in accidents at Monza, some on or approaching the parabolica including Wolfgang (Taffy) von Trips (1928–1961) and Jochen Rindt (1942-1970).  In 2021, the Monza authorities announced the parabolica officially would be renamed “Curva in honor of former Ferrari factory driver Michele Alboreto (1956-2001) who to date remains the last Italian driver to win a Formula One Grand Prix for Scuderia Ferrari.  It’s likely most will still refer to the curve as “the parabolica”.

The Monza circuit in its configuration for the 1955 Italian Grand Prix (left) and a Mercedes-Benz W196R (streamliner) exiting the parabolica ahead of two W196Rs in conventional open-wheel configuration.  The 1955 Italian Grand Prix was the seventh and final round of the World Championship of Drivers, the French, German, Swiss and Spanish events all cancelled in the aftermath of the disaster at Le Mans.  It was the fourth and last appearance of the Mercedes-Benz W196R streamliners which, after some bad experiences on the relatively tight Silverstone circuit, were restricted to the fast, open tracks.  Mercedes-Benz also withdrew from top-level competition after 1955 and, as a constructor, it would be half a century before they returned to Grand Prix racing.

The parabolic arc: A wheel drops off a Boeing Dreamlifter on take-off, describing a a classic parabolic arc.  The Boeing 747-400 Large Cargo Freighters (LCF) were created using a modified 747-400 airline frame and were most associated with their use carrying Boeing 787 Dreamliner parts between the US, Italy & Japan.  It was an unusual configuration in that it was required to carry components which while large, weren't particularly heavy.

Perimeter & Parameter

Perimeter (pronounced puh-rim-i-ter)

(1) A line bounding or marking off an area; any boundary around.

(2) The outermost limits.

(3) In geometry, the border or outer boundary of a two-dimensional figure (the sum of the lengths of the segments that form the sides of a polygon.

(4) The total length of such line; the total length of any such closed curve, such as the circumference of a circle.

(5) In military jargon a fortified boundary that protects a position.

(6) In clinical ophthalmology, an instrument for determining the peripheral field of vision.

(7) In basketball, a semicircular line on a basketball court surrounding the basket, outside of which field goals are worth three points rather than two (also called three-point line).

(8) The area outside this line (often used attributively).

1585–1595: From the French périmètre (circumference, outer boundary, or border of a figure or surface), from the feminine Latin form perimetros, from the neuter Greek perímetron (circumference), the construct being peri- (around; about) + -meter from metron (measure), from the primitive Indo-European root me- (to measure).  The military sense of “boundary of a defended position” is said by some sources to have come into use only by 1943 despite the tactic being probably the second oldest military procedure still in use (the attack presumably the first).  Whether coincidental or not, the ultimate failure of perimeter defense was what finally led to the success of the Soviet offensive against the Nazi Sixth Army in Stalingrad (now Volgagrad) in 1943.  The technical terms created by the use of perimeter as a modifier include perimeter check (a patrol which checks to ensure a defensive perimeter remains in place) & perimeter fence.  Perimeter & perimetry are nouns, perimetral, perimetric & perimetrical are adjectives and perimetrically is an adverb; the noun plural is perimeters.

Parameter (pronounced puh-ram-uh-tuhr (U) or puh-ram-i-ter (non-U)

(1) In mathematics, a constant or variable term in a function that determines the specific form of the function but not its general nature, as a in f(x) = ax, where a determines only the slope of the line described by f(x).  (A value kept constant during an experiment, equation, calculation or similar, but varied over other versions of the experiment, equation, calculation etc).

(2) In mathematics, one of the independent variables in a set of parametric equations.

(3) In geometry, in the ellipse and hyperbola, a third proportional to any diameter and its conjugate, or in the parabola, to any abscissa and the corresponding ordinate.

(4) In crystallography, the ratio of the three crystallographic axes which determines the position of any plane; the fundamental axial ratio for a given species.

(5) In statistics, a variable entering into the mathematical form of any distribution such that the possible values of the variable correspond to different distributions (any measured quantity of a statistical population that summarizes or describes an aspect of the population).

(6) In computing, a variable that must be given a specific value during the execution of a program or of a procedure within a program.

(7) Limits or boundaries; guidelines; specifications; any constant, definitional or limiting factor (usually in the plural parameters).

(8) Characteristic or a factor; an aspect or element.

(9) In computing syntax for various purposes, an input variable of a function definition, that become an actual value (argument) at execution time (an actual value given to such a formal parameter).

1650-1660: From the French paramètre, from the New Latin parametrum (parameter), the construct being the Ancient Greek παρα- (para-) (beside, subsidiary) + μέτρον (métron) (meter) (measure), from the primitive Indo-European root me- (to measure).  The words was almost exclusive to mathematics & geometry until the late 1920s when it came to be extended to “measurable factor(s) which help to define a particular system", hence the now common alternative meaning “boundary, limit, characteristic factor” (under the influence of perimeter which used a similar spelling and (at least conceptually) could be understood to enjoy some overlap of meaning.  Although the wider definition has been in use since the 1950s, purists have never approved.  Parameter is a noun and parametric & parametrical are adjectives; the noun plural is parameters.

Parameters and perimeters

The more modern ways “parameter” has been used since the early twentieth century does offend the linguistically more fastidious but it seems clear the innovations are here to stay.  Some do however just get it wrong and university lecturers in the social sciences seem to be those who bear the heaviest burden of training a certain number of their institution’s first year students in the correct use of “parameter” & “perimeter”.  That they are sometimes confused is understandable because the spellings are so close and there is some sense of overlap in the meanings, both able to be used in a way which defines limits.  The definitions can be reduced to: (1) perimeter refers to either something physical (a national border; a fence etc) or a representation of something physical (lines on a map; the four sides of a square etc) whereas (2) a parameter is an element of specification, a constant or variable value which can be either an absolute value or a range.  So, a perimeter may be drawn on the basis of certain parameters while the values of parameters will in some cases exist within certain perimeters.  Definitions such as that are vague enough for those so inclined to find contradictions but for the way most people, most of the time (correctly) use parameter & perimeter, it seems serviceable.

Lindsay Lohan and her lawyer in court, Los Angeles, December 2011.