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tigatprt for an Ulcer he had on his Leg, but which the Add 4. A very nobleTheorem, in Elementary Genm,*™
People conftrued a Diadem. , „ »™ demonftrated by Mr. Z<Mj>, in the/Jfa,,;/ v. 7' 5
4liny Lib. VII. C. ;. obferves, thatEacchus was the firft demie Royale des Sciences, An.i-jotS. That tfe Sunofti
Inventor of the Diadem: Athetunts affuresu?, that it was Squares of the two Diagonals of every 'ParalkCrai,-,
the Topers, and good Fellows, who firft made Ufeofit, to equal to the Sum of the Squares of the four sides preferce themfelves from the Fumes of Wine, by tying it 'Tis evident, at firft Sight, that the famed 47'th p mn „
fight round their Heads ; and that it afterwards came to fition of Euclid, fo richly worth the Hecatomb it coll V
be a Royal Ornament. , . „ _ , ,; Author, is 1 only a particular Cafe of this Proposition :» *
The Diadem remain'd a long Time the peculiar Badge ot it the Parallelogram be redangled, it follows of Courfe th '-
Kings- At length it was affirmed by the Roman Emperors, the two Diagonals are equal ; and, of Confequence' that
as the Mark of Empire. the Square of a Diagonal, or, which is the lame Thing,
Authors are not agreed about the Time when tmRoman the Square ot the Hypothenule of a right Angle, is equal to
Emperors firft affiim'd theDiadem. Some refer it to Co- the Squares of the two Sides. If a Parallelogram be obliq Ue .
Tin/la others to Aurelian, and others to Confiantine the angled, and of Confequence, the two Diagonals unequal
Great. The younger Vitlor fays positively, that Aurelian as is the more ufual Cafe 5 the Propofition becomes of more
took the Diadem, which no Emperor had dared to do be fore him. For tho' it /hould feem from the fame Writer, that Caligula had done the like, yet Suetonius allures us, he had it only in View, and that he never executed it. Heliogabalus, indeed, took in the Palace he wore
extenfive Ufe.
The Demonftration in oblique-angled Parallelograms
is thus : Suppofe the oblique-angled Parallelogram
ABCD, (Tab. Geometry Fig. 25.) whereof ED is the
Diadem, but it was only greater Diagonal, and AC the k-ffer : From the Point A,
and never appear'd with of the obtufe Angle DAB, let fall a Perpendicular AE,
it in publick. Jornandus even goes as low as Diode- the Side CD ; and from the Point B another Perpendicular film for the ihtroduflion of the Diadem ; Bat 'tis BF to the Side DC. Then are the Triangles ADE, BCF certain there is a Medal of Aurelian, with a Crown equal, and fimilar, as AD is equal to BC, and the Angles like one of our Ducal Crowns, which is fuitain'd by aBor- ADE, BCF, as well as AED, BCF, are alfo equal ; con- dor of Pearls, that bears a very great Affinity to zDiadem. fequently DE is equal to CF. Now, by Euclid, Prop. 12. And the Authors, who have explain'd that Medal, are all Lib. II. in the obtufc-angled Triangle BDC, the Square of agreed it is one. Mr. Spanheim alfo allows Aurelian to the Side BD is equal to the Sum of the Squares of BC, and have taken it: His Succeffors imitated him therein ; And CD, and over and above, to double the Rectangle ofCE yet theOrnament did not become common till the Time of by CD ; and by 13th, Lib. II. In the Triangle DAC, the Confiantine. After him the Empreffes were allowed to Square of the Side AC is equal to the Sum of the Squares wear it: accordingly we find them reprefentcd therewith on of AD, and CD, abating double the Rectangle of the fame Medals: Tho' till'then, we have no Inftance either of Crown, CD, by DE, equal to CF. Confequently, the former Ex- or Diadem on a Woman's Head, in all the Roman Empire. C efs precifely compenfating this Defect ; the Sum of the
An Author of the Vth Century quoted by Bollandus, Squares of the two Diagonals is equal to the Sum of the
pretends, that Confiantine firft wore theDiadem, and that Squares of tho four Sides QED.
he only took it to bind his Hair, and keep it in Order. But Hence, In every Rhombus, or Lozange, knowing one
this is not very probable; and 'tis certain, that at leaft Side, and a Diagonal ; the other Diagonal will likewife
feme Emperors had wore it before him, as Aurelius, and be knotfn : For as the lour Sides are equal; fubftracting
Carimts. Eufebius attributes it to Conflantius Chlorus, the Square of the given Diagonal from Quadrunle the
when only Crefar^ which is confirm'd by one of his Medals, Square of the given Side ; the Remainder is the 'Square
wherein he is reprefented with a Diadem, adorn'd with of the Diagonal required.
Rays; Tho' even after Confiantine, when the Diadem was The Propofition is likewife of great Ufe in the Theory of become the ufual Ornament of the Augnfti, it was not al- compound Motions : For in an oblique-angled Parallelo- ways given to the Ctefars. Indeed, we fee it on feme of gram, the greater Diagonal being the Subtenfe of an ob- the Medals of Julian, while only Ctffar ; tho' 'tis pretty tufe; and the leffer, of an acute Angle, which is the Corn- certain, he did not wear it till he became Augufius. Du plement of the former ; the greater will be the greater, and Cango will not maintain, that Confiantine firft took the the lefs, the lefs, as the obtufe Angle is the greater : So Diadem; but only, that he firft made it into a Kind of that if the obtufe Angle be conceiv'd to grow till it be in- Calk, or Clofe-Crown, as is feen in fomc of his Medals, and finitely great with regard to the acute one, or, which thofe of his Succeffors. amounts to the fame Thing, if the two contiguous Sides of
The Word comes from the Latin Diadema, of the Greek the Parallelogram be extended directly, End to End in a
fixity.*, a little 'Sand encompaffing the Head, of tho Verb right Line ; the great Diagonal becomes the Sum of the
JW4», Alligo, I tye. two Sides, and the leffer one, nothing. Now, two conti-
Diadem, in Heraldry, is applied to certain Circles, or guousSides of a Parallelogram being known, together with
Rims fcrvi'ng to bind, or inclofe the Crowns of Soveraign the Angle they include, 'tis eafy to find, the Subtenfe of
Princes • and to bear the Globe, and Crofs, or the Flower de that Angle, i. e. one of the Diagonals of tho Parallelogram,
Lis for 'their Crcft. in Numbers; which done, Mr. de Zagny's Propofition gives
The Crowns of Sovereigns differ in this, that fome are the other. Which fecond Diagonal thus found, is the Line
bound with a greater, and fome with a lefs Number of that would be defcribed by a Body impcll'd at the fam»
Diadems. Time by two Forces, which Ihould have the fame Ratio
Prelates likewife appear to have anciently wore a Sort of to each other, as the contiguous Sides have, and aft on
Diadem : Thus Saronius writes, that St. James the thofe two Direflions ; which Diagonal, the Body would
Apoftle wore a <rold Plate on his Fore-head, as a Mark of defcribe in the fame Time, as it would have defcribed
his Epifcopal Di?nity. eitner of the contiguous Sides in, if only impell'd by
In Blazoning the Bandage about the Heads of Moors, the Force correfpondmg thereto. This is one of the great
on Shields, is fometimes alfo call'd Diadem. Ufes of the Propofition : For the Ratio of two Forces, and
DIAGLYPH1CE, the Art of Engraving, Cutting, or the Angle they make, being given, it is frequently neceffa-
othcrwife working hallow, or concave Figures, in Metals; ry to determine, in Numbers, the Line a Body impcll'd by
u ~ in a certain Time. See-
the two Forces would defcribe Compound Motion.
AH the Sides of a re£ti-]inear Figure, as AB, BC, CD, DE, (Fig-iS.) excepting one EA, and the Angles O, and T, be- ing given ; to find the Diagonals.
In the Triangle ABE, the Sides AB, and AE being given, the Angle O is eafily found by Trigonometry ; and
Such as Seals Intaglia's, the Matrices, or Coins for Me- dals, &c. See Engraving, and Sculpture.
DIAGNOSTIC, in Medicine, a Term applied to thofe Signs, or Symptoms, which indicate, or difcover the pre- fent State of a Difeafe, itsNature, and Caufe. See Sign, and Indication.
The Phyficians have Diagnofiic, and c Prognofiic Signs ; the firft with regard to the prefent State of the Difeafe, from this, the Diagonal BS. And after the like Manner and the Patient ; and the fecond to the future. See Pro- the Triangle BCD is refolved, and the Diagonal BD found. cnostic. Since Iconographies, or Plans, are beft raken by having
The Word is compounded of the Greek fit!, per, through, all the Sides, and Diagonals ; The Ufe of this Problem in or by; and yUw*.u 7 Ihioiv. Planimetry is of fome Importance ; efpecially to fuch as
DIAGONAL, in Geometry, a right Line drawn a-crofs are willing to have their Work accurate, tho' at the Ex- Figure of feveral Sides, from the Vertex of one Angle to pence of Calculation. See Ichnographv, &c,
a Figure or lcvcrai ojucb, irum rne vertex of one Angl- that of another : Such is the Line PN" (ftab.Gcometr. Fig. 24. ) drawn from the Angle P, to N. See Figure.
Some Authors call it Diameter, and others Diametral of the Figure. See Diameter.
It is demonftrated, 1. That every Diagonal divides a Pa- rallelogram into two equal Parts. 2. Two Diagonals drawn in any Parallelogram, biffect each other. 3. The Diagonal of a Square is incommenfurable with one of its Sides, See Parallelogram, Sqjjare, i$c.
nocraphy,
DIAGRAM, in the ancient Mufic, was what we call the Scale, orGammut in the modern. SecScALE.and Gammut.
The Extent of the Diagramma, which they alfo call'd, Syfiema perfetlum, was a Difdiafafon, or two Octaves in the Ratio 1:4. In that Space they had eighteen Chords, tho' thefe had not all different Sounds. See Chord.
To explain it, they reprefent to us eighteen Chords, or Strings of an Inftrument, as the Lyre, fuppofed to be tu- ned according to the Proportions in any of the Genera, viz.
Dia-
