CARRIED DIAGONALS. FROM THE CENTRE INSCRIBE A SQUARE. THE INTERSECTIONS
OF THE LINES OF THE STAR COMPLETE THE FIGURE.]
[Illustration: 51 DESCRIBE CIRCUMFERENCES EQUAL AND TANGENT, AND DIVIDE
INTO SIXTEEN EQUAL PARTS. BY THE ANGLES OF DIVISION DESCRIBE A PENTAGON
STARRED. BY THE CENTRE OF THE SQUARE AN OCTAGON STARRED, FROM WHICH
EMANATES AN OCTAGON ROSETTE. HEXAGONS REGULAR AND STARRED.]
[Illustration: 52 DIVIDE TANGENT CIRCUMFERENCES INTO SIXTEEN EQUAL
PARTS. SIMILAR DISPOSITION TO FIGURE NUMBER 49, BUT WITH DIFFERENT
TREATMENT.]
[Illustration: 53 DESCRIBE CIRCUMFERENCES AS IN PLAN; INSCRIBE THEREIN
BY THE DIAGONALS A STAR, THE SIDES OF WHICH, PROLONGED AND INTERSECTED
BY THE OCTAGONAL STAR, DETERMINE THE ROSETTES.]
[Illustration: 54 DIVIDE CIRCUMFERENCES AS IN PLAN INTO THIRTY-TWO EQUAL
PARTS. FROM THE CENTRE OF THE SQUARE INSCRIBE A STAR OF SIXTEEN POINTS,
THE PROLONGATION OF ITS LINES FORMING THE ROSETTE OF SIXTEEN.]
[Illustration: 55 DESCRIBE CIRCUMFERENCES AS IN PLAN. FROM THE CENTRE OF
THE PLAN BY RADIATING LINES INSCRIBE A STARRED OCTAGON; THE PROLONGATION
AND MEETING OF ITS LINES IN REPETITION COMPLETE THE FIGURE.]
[Illustration: 56 DESCRIBE CIRCUMFERENCES AS IN PLAN. IN THE CENTRE OF
FOUR EQUAL SQUARES TRACE AN OCTAGONAL ROSETTE, AFTER HAVING TAKEN IN THE
LARGE ROSETTE OF SIXTEEN POINTS, WHICH LENDS ITS LINES TO THE FORMATION
OF EIGHT SURROUNDING CRUCIFORM FIGURES.]
[Illustration: PLATE LXXVIII.
SEVILLE.
Socle of the Entrance Arch to the Chapel.]
[Illustration: 57 FROM THE CENTRE OF FOUR EQUAL SQUARES DESCRIBE
CIRCUMFERENCES AS IN PLAN. INSCRIBE THEREIN THE STARRED OCTAGON, THE
PROLONGED SIDES OF WHICH DETERMINE THE QUADRILATERAL ROSETTE.]
[Illustration: 58 THE SQUARE OF DISTRIBUTION IS A RECTANGLE LENGTHENED,
FROM WHICH OCTAGONS ARE TRACED. TERNARY STARS, YET WITH SIX POINTS, AND
PAIRED STARS WITH FIVE POINTS FILL IN THE RECTANGLE.]
[Illustration: 59 DESIGN OF FOUR FIGURES. AN OCTAGON, A HEXAGON PAIRED,
A STARRED PENTAGON, AND A STARRED HEXAGON.]
[Illustration: 60 ANALOGOUS TO NUMBER 58, WITH CLOSER DEVELOPMENT.]
[Illustration: 61 A STARRED OCTAGON, THE PROLONGED LINES OF WHICH FORM
AN OCTAGONAL ROSETTE, SEPARATED BY A REGULAR HEXAGON.]
[Illustration: 62 DIVIDE CIRCUMFERENCES AS IN PLAN INTO SIXTEEN EQUAL
PARTS. THE DIAGONALS WILL GIVE A STAR OF SIXTEEN POINTS, THE LINES OF
WHICH, EXTENDED, FORM A ROSETTE OF SIXTEEN POINTS WITHIN A SQUARE. THE
ANGLES OF THE SQUARE INTERSECT REGULAR HEXAGONS.]
[Illustration: 63 DESCRIBE THE CIRCUMFERENCES AS IN THE PLAN. INSCRIBE
FROM A CENTRE A STARRED OCTAGON ENCLOSED WITHIN A REGULAR OCTAGON, A
STARRED HEXAGON WITHIN ALTERNATE HEXAGONS, AND A CRUCIFORM FIGURE WITHIN
A FOUR-POINTED STAR.]
[Illustration: 63’ DESCRIBE CIRCUMFERENCES AS IN THE PLAN, AND FROM A
CENTRE INSCRIBE A STARRED OCTAGON; FROM THE EXTENDED LINES IS FORMED A
CRUCIFORM FIGURE. FROM OTHER CENTRES INSCRIBE STARRED AND REGULAR
HEXAGONS.]
[Illustration: SEVILLE.
Detail of the Tiles of the Altar.]
[Illustration: 64 DESCRIBE CIRCUMFERENCES AS IN THE PLAN. FROM A CENTRE
INSCRIBE A STARRED OCTAGON OF WHICH THE SIDES ARE PROLONGED. BY THESE
PROLONGATIONS, AND BY OCTAGONAL FIGURES IN PAIRS, THE TRACING IS
COMPLETE.]
[Illustration: 65 DIVIDE CIRCUMFERENCES AS IN THE PLAN INTO TWENTY-FOUR
EQUAL PARTS, AND INSCRIBE FROM A CENTRE A STARRED DODECAGON, THE
EXTENDED LINES OF WHICH DETERMINE THE INTERSECTING LINES OF THE
ROSETTE.]
[Illustration: 66 DESCRIBE CIRCUMFERENCES AS IN THE PLAN. FROM A CENTRE
INSCRIBE A STARRED DODECAGON ANALOGOUS TO NUMBER 65.]
[Illustration: 67 TRIANGULAR PLAN. DESCRIBE CIRCUMFERENCES AS IN THE
PLAN. FROM A CENTRE INSCRIBE A STARRED DODECAGON, THE EXTENDED LINES OF
WHICH FORM THE UNEQUAL LIMBS OF A ROSETTE, AND A CRUCIFORM FIGURE WITHIN
A SQUARE.]
[Illustration: PLATE LXXX.
SEVILLE.
Socle in the Interior of the Chapel.]
[Illustration: 68 DIVIDE AS IN THE PLAN. INSCRIBE A STARRED DODECAGON,
THE SIDES OF WHICH PROLONGED INSCRIBE THE LINES OF THE ROSETTE. FOUR
ROSETTES PENETRATE EACH OTHER, AND ARE EACH INVADED BY A STAR HAVING
TRIANGULAR WEBS.]
[Illustration: 69 FROM A CENTRE AS IN THE PLAN DESCRIBE A STARRED
DODECAGON, THE SIDES OF WHICH PROLONGED FORM THE LINES OF THE ROSETTE.
THE ROSETTE, BY EXTENDING LINES, IS SURROUNDED BY TWELVE HEXAGONS.]
[Illustration: 70 DIVIDE AS IN THE PLAN. DESIGN ANALOGOUS TO NUMBER 69.
BY EXTENSION OF LINES OF THE ROSETTE HEXAGONS ARE GROUPED.]
[Illustration: 71 DIVIDE AS IN PLAN. DESIGN ANALOGOUS TO NUMBER 70.]
[Illustration: PLATE LXXXI.
SEVILLE.
Socle in the Interior of the Chapel.]
[Illustration: 72 DIVIDE AS IN PLAN. THE CIRCUMFERENCES DIVIDED INTO
TWENTY-FOUR EQUAL PARTS DETERMINE THE ROSETTE, THE EXTENDED LINES OF
WHICH DESCRIBE THE STARRED PENTAGON.]
[Illustration: 73 CURVILINEAR TRANSFORMATION OF NUMBER 72 BY THE
SUBSTITUTION OF ARCS FOR RECTILINEAL FEATURES.]
[Illustration: 74 DIVIDE THE CIRCUMFERENCES INTO TWENTY-FOUR EQUAL
PARTS. INSCRIBE THE STARRED DODECAGON, THE EXTENDED LINES OF WHICH
DESCRIBE THE ROSETTE.]
[Illustration: 75 DESCRIBE THREE CONCENTRIC CIRCUMFERENCES AS IN PLAN.
THE ROSETTE BECOMES ENTIRE BY LINES EXTENDED FROM THE STARRED DODECAGON.
CROSSED LINES FROM THE ROSETTE DETERMINE THE SQUARE.]
[Illustration: 76 TRANSFORMATION OF NUMBER 75. ROSETTE IDENTICAL. IN THE
CENTRE OF A SQUARE DESCRIBE AN OCTAGON, THE PROLONGED SIDES OF WHICH
INVADE THE SQUARE WHICH FIGURES AROUND THE ROSETTE.]
[Illustration: 77 DIVIDE AS IN PLAN. FROM A CENTRE TRACE A STARRED
HEXAGON, THE EXTENDED LINES OF WHICH CUT THE PROLONGED LINES OF THE
ROSETTE.]
[Illustration: 78 DESCRIBE CIRCUMFERENCES AS IN PLAN. FROM A CENTRE
TRACE A STARRED DODECAGON. THUS ARE DETERMINED ROSETTES PLACED END TO
END, EACH IN A REGULAR HEXAGON.]
[Illustration: 79 DESCRIBE CIRCUMFERENCES AS IN PLAN. THE ROSETTE IS
DETERMINED BY THE STARRED DODECAGON. THIS DESIGN IS MOST DIVERSIFIED,
AND CAPABLE OF EXHAUSTIVE FORMS OF ORNAMENT.]
[Illustration: 80 FROM A CENTRE DRAW A STARRED DODECAGON, WHICH
DETERMINES THE ROSETTE, THE CROSSED LINES AT THE POINTS OF THE ROSETTE
DETERMINING THE MANY REGULAR HEXAGONS AND _tricèles_.]
[Illustration: 81 HEXAGONAL DISTRIBUTION. FROM A CENTRE DRAW A STARRED
DODECAGON DETERMINING THE ROSETTE, THE ALTERNATELY CROSSED LINES OF
WHICH FORM A _tricèle_, WITHIN A SECOND CIRCUMFERENCE, AS IN THE PLAN,
DRAW A STARRED HEXAGON, THE EXTENDED LINES FORMING SIX REGULAR
HEXAGONS.]
[Illustration: 82 FROM A CENTRE A STARRED DODECAGON. THE LINES EXTENDED
FORM AN OUTER STARRED DODECAGON, AND BY CROSSING DESCRIBES A STARRED
HEXAGON AND A LOZENGE, WITHIN WHICH IS A CRUCIFORM FIGURE.]
[Illustration: 82’ THE RADIUS OF THE CIRCUMFERENCE IS EQUAL TO A THIRD
OF THE HEIGHT OF THE TRIANGLE, AND THE SQUARE MAKES A STAR OF FOUR
POINTS AT EACH ANGLE.]
[Illustration: 83 DRAW CIRCUMFERENCES AND DIVIDE AS IN PLAN. THE STARRED
DODECAGON DETERMINED BY INTERTWINED SQUARES. THE LINES OF THE DODECAGON,
EXTENDED AT INTERVALS, FORM A STAR OF FOUR POINTS.]
[Illustration: 84 DESCRIBE CIRCUMFERENCES AS IN PLAN. FROM A CENTRE AN
INNER AND OUTER STARRED DODECAGON, THE LINES OF WHICH EXTENDED FORM A
STARRED AND REGULAR HEXAGON, INCLUDING A STARRED OCTAGON WHICH MERGES
INTO A CRUCIFORM FIGURE.]
[Illustration: 85 TRACE THE NET-WORK OF THE DODECAGON, THE HEXAGON, AND
THE FOUR-POINTED STARS. FOLLOW DIAGONALS AS IN PLAN.]
[Illustration: 86 DRAW CIRCUMFERENCES AS IN PLAN. FROM A CENTRE AN INNER
AND OUTER STARRED DODECAGON. EXTENDED LINES OF THE INNER DODECAGON FORM
SIX SQUARES WHICH INVADE A REGULAR DODECAGON.]
[Illustration: Mosaics from Various Halls.]
[Illustration: 87 SQUARE PLAN. AN INNER AND OUTER STARRED DODECAGON, AND
A REGULAR DODECAGON BY DIAGONALS.]
[Illustration: 88 SQUARE PLAN. DESCRIBE CIRCUMFERENCES AND DIVIDE THEM
INTO TWENTY-FOUR EQUAL PARTS, AND DRAW THE DIAGONALS OF EIGHT IN EIGHT
DIVISIONS. THE REST FOLLOWS.]
[Illustration: 89 NET-WORK OF THE DODECAGON AND THE TRIANGLE ASSEMBLED.
BY EXTENDED LINES OF THE STARRED DODECAGON AN IRREGULAR POLYGON, AND A
ROSETTE OF TWELVE LIMBS ARE FORMED.]
[Illustration: 90 ISOCELES PLAN. A CIRCUMFERENCE IS DRAWN IN A SQUARE
AND DIVIDED INTO TWENTY-FOUR EQUAL PARTS. A CIRCUMFERENCE, CONCENTRIC TO
THE FIRST, COMPLETES THE ROSETTE BY MEANS OF DIAGONALS. THE SMALL
HEXAGON AND THE OCTAGON ARE TRACED.]
[Illustration: 91 DESCRIBE CIRCUMFERENCE AND DIVIDE INTO TWENTY-FOUR
EQUAL PARTS. THE REGULAR DODECAGON IS DRAWN. THE INNER AND OUTER STARRED
DODECAGONS ARE DESCRIBED BY RADIATING LINES.]
[Illustration: 92 NET-WORK OF THE HEXAGON AND THE TRIANGLE. THE ROSETTE
OF TWELVE POINTS SPRINGING FROM A STARRED DODECAGON IS ENCLOSED BY A
HEXAGON, TRELLISED, FROM WHICH THE _tricèles_ ARE DRAWN.]
[Illustration: 93 OF SIMILAR INTENTION TO NUMBER 92, BUT AN IRREGULAR
HEXAGON RECEIVES INTERCALARY LOZENGES.]
[Illustration: 94 TRIANGULAR PLAN. TRACED BY TRELLISED NET-WORK. THE
HEXAGON ENVELOPED IN SPIRALS.]
[Illustration: 95 TRIANGULAR PLAN. HEXAGON ENVELOPED IN SPIRALS.
ANALOGOUS TO NUMBER 94.]
[Illustration: 96 NET-WORK OF THE OCTAGON, HEXAGON, AND CIRCLE,
ASSEMBLED. FROM THE STARRED OCTAGONS A CURVILINEAL ROSETTE.]
[Illustration: 97 SQUARE PLAN. STARS AND ROSETTES. DESCRIBE
CIRCUMFERENCES AS INDICATED. THE OCTAGONAL STARS RECEIVE THE EXTENDED
LINES OF THE HEXAGONAL AND PENTAGONAL STARS. THE REST FOLLOWS.]
[Illustration: 98 DIVIDE CIRCUMFERENCES AS IN THE PLAN. FROM THE CENTRE
OF THE HEXAGONAL ROSETTE DESCRIBE A CIRCUMFERENCE TANGENT TO THE FIRST,
AND DIVIDE INTO TWELVE PARTS. BY THE AID OF THE PENTAGON COMPRISED
COMPLETE THE ROSETTE; THEN, DEPENDING ON THE PENTAGON--WHICH, THOUGH
IRREGULAR, RULES ALL--TRACE THE PENTAGONAL FIGURE WHICH STANDS ON THE
POINTS OF THE ROSETTES.]
[Illustration: PLATE LXXXIII.
Mosaics from Various Halls.]
[Illustration: 99 ISOCELES PLAN. DIVIDE THE SPACE SURROUNDING THE ACUTE
APICES OF THE LOZENGE INTO TWENTY EQUAL PARTS, AND OF THE OBTUSE APICES
INTO SIXTEEN. IN THE ONE DRAW A STARRED DECAGONAL, AND IN THE OTHER A
STARRED OCTAGONAL ROSETTE. THE ADJUSTMENT OF TWO TYPES SO DIFFERENT IS
NOT CONSIDERED SUCCESSFUL.]
[Illustration: 100 DIAGRAM OF THE DODECAGON, HEXAGON AND SQUARE. IN THE
CENTRE OF THE SQUARE A REGULAR OCTAGON, WHICH IS SUPREME. THEN FROM THE
STARRED DODECAGON INSCRIBE A ROSETTE OF TWELVE POINTS; LASTLY, FROM THE
STARRED HEXAGON INSCRIBE A ROSETTE OF SIX POINTS.]
[Illustration: 101 DIVIDE THE CIRCUMFERENCES AS IN THE PLAN, THE ONE
INTO TWENTY-FOUR AND THE OTHER INTO TWELVE EQUAL PARTS. FOR THE REST,
THE CENTRES BEING INDICATED, IT IS EASY TO TRACE THE ARCS, AND SO
COMPLETE THE FIGURE.]
[Illustration: 102 THE LOZENGE AND TRELLISED ROSETTE EMANATING FROM
STARRED HEXAGON, ALTERNATING ROW BY ROW WITH DODECAGONAL ROSETTE
SPRINGING FROM STARRED DODECAGON.]
[Illustration: PLATE LXXXIV.
Part of Ceiling of a Portico.]
[Illustration: 103 DESCRIBE THE CIRCUMFERENCES AS IN PLAN. BY THE POINTS
OF THE STARRED HEXAGON DESCRIBE A REGULAR HEXAGON. BY LINES EXTENDED
FROM THE STARRED HEXAGON THE TWELVE-POINTED ROSETTE IS FORMED,
CONSTRUCTING AT THE SAME TIME THE STARRED DODECAGON.]
[Illustration: 104 DESIGN ANALOGOUS TO NUMBER 103, BUT WITH CHANGE OF
_motif_.]
[Illustration: 105 SQUARE PLAN. DIVIDE AS INDICATED AND INSCRIBE A
STARRED DODECAGON, THE EXTENDED LINES OF WHICH FORM AN OUTER STAR ALSO
OF TWELVE POINTS; THE STARRED OCTAGON IS CONSTITUTED BY POINTS PROLONGED
FROM THE DODECAGON.]
[Illustration: 106 SQUARE PLAN. SIMILAR GROUND-WORK TO THAT OF NUMBER
105. THE DODECAGONAL AND OCTAGONAL ROSETTES ARE DESCRIBED BY CONCENTRIC
CIRCUMFERENCES.]
[Illustration: 107 SQUARE PLAN. SIMILAR GROUND-WORK TO NUMBERS 105 AND
106. THE TREATMENT CONSIDERABLY CHANGED.]
[Illustration: 108 SQUARE PLAN. SUB-DIVIDE AS INDICATED. INSCRIBE THE
PENTAGON, THE EXTENDED LINES OF WHICH ESTABLISH THE OCTAGONAL STAR AND
ROSETTE, AS WELL AS THE DODECAGONAL ROSETTE AND STAR.]
[Illustration: 109 SQUARE PLAN. DIVIDE AS INDICATED. THE FLOWER-WORK
WHICH ACCOMPANIES THE IRREGULAR OCTAGON, THOUGH INDEPENDENT OF GEOMETRIC
CONSTRUCTION, IS YET WITHIN THE PROPULSION OF THE LINES.]
[Illustration: 110 GROUND-WORK ANALOGOUS TO NUMBER 108, BUT ON A
TRIANGULAR PLAN. THE EXTENDED LINES OF THE PENTAGON GOVERN THE
ENNEAGONAL AND DODECAGONAL STAR AND ROSETTE.]
[Illustration: 111 CURVILINEAR TRANSFORMATION OF NUMBER 110.]
[Illustration: 112 TRIANGULAR PLAN. DISTRIBUTION OF ENNEAGONAL AND
DODECAGONAL STARS AND ROSETTES. [THE DODECAGON ONLY PARTIALLY
DISPLAYED.]]
[Illustration: 113 TRIANGULAR PLAN. SIMILAR CONSTRUCTION TO NUMBER 112.
THE DODECAGONAL STAR, ROSETTE, AND OUTER STAR, WHICH ARE IN THE DIAGRAM
NUMBER 112 ONLY PARTIALLY SEEN, ARE HERE DISPLAYED.]
[Illustration: 114 TRIANGULAR PLAN. FROM A CENTRE AS IN PLAN DESCRIBE
CIRCUMFERENCES, WHICH RULE THE DODECAGONAL STAR, ROSETTE, AND OUTER
STAR. MANY OF THE FIGURES ARE DRAWN INDEPENDENTLY, THOUGH GOVERNED BY
DIVISIONS.]
[Illustration: 115 DIVIDE AS IN PLAN. INSCRIBE A DECAGONAL ROSETTE, FROM
THE PROLONGED LINES OF WHICH PROCEED THE PENTAGONAL STAR, THE REGULAR
OCTAGON, AND OTHER FIGURES.]
[Illustration: 116 DIVIDE AS IN PLAN. TWO DIFFERENT ROSETTES, ONE OF
TWELVE AND THE OTHER OF FIFTEEN POINTS. [THE DODECAGONAL ROSETTE IS,
HOWEVER, ONLY PARTIALLY SHOWN HERE.]]
[Illustration: 117 DESCRIBE CIRCUMFERENCES AND TRACE ROSETTE OF SIXTEEN
POINTS. THE LINES EXTENDED WILL COMPLETE THE FIGURE.]
[Illustration: 118 CURVILINEAR TRANSFORMATION OF NUMBER 117.]
[Illustration: 119 DIVIDE AS INDICATED, AND THE RESULTING ROSETTE OF
SIXTEEN POINTS, WHICH WOULD NATURALLY BE RECTILINEAR, MAY BE EASILY
TRANSFORMED TO CURVILINEAR; WHILE THE PENTAGONAL STARS, TREATED IN
UNDULATING FORM, BECOME FLOWER-WORK OR FOLIAGE.]
[Illustration: 120 SAME GROUND-WORK AS NUMBER 119. BUT HERE THE ROSETTE
IS STARRED, END ON END, ABOUT THE POINTS.]
[Illustration: 121 SUB-DIVIDE AS IN PLAN. THE CIRCUMFERENCE GOVERNING
THE HEPTAGON IS DIVIDED INTO EQUAL PARTS; BY PROLONGING THE SIDES OF THE
HEPTAGON THE ROSETTE OF SIXTEEN POINTS IS INSCRIBED.]
[Illustration: 122 CIRCUMFERENCES TANGENT TO THOSE OF THE PENTAGON
INSCRIBE THE ROSETTE OF SIXTEEN POINTS.]
[Illustration: 123 TRANSFORMATION OF THE RECTILINEAL ROSETTE NUMBER 122.
THE FIGURES IN OTHER RESPECTS IDENTICAL.]
[Illustration: 124 DIVIDE AS INDICATED. THE EXTENDED LINES OF THE
HEXAGON INSCRIBE THE ROSETTE OF EIGHTEEN POINTS AND THAT OF NINE POINTS.
THE REST IS BUT A MATTER OF ADJUSTMENT.]
[Illustration: 125 DESIGN ANALOGOUS TO NUMBER 124. IN THIS ARRANGEMENT,
TWO SIDES OF THE HEPTAGON PROLONGED DETERMINE THE ROSETTE OF EIGHTEEN
AND THAT OF TWELVE POINTS.]
[Illustration: 126 SQUARE PLAN. DIVIDE AS INDICATED. INSCRIBE A ROSETTE
OF TWENTY POINTS (THE HALF OF WHICH IS SHOWN IN THE DIAGRAM). IN A
TANGENT CIRCUMFERENCE INSCRIBE THE ROSETTE OF TWELVE POINTS. THE REST
FOLLOWS.]
[Illustration: 127 DISTRIBUTION OF THE DODECAGON, HEXAGON, AND SQUARE
ASSEMBLED. BY THE MIDDLE POINTS OF THE SIDES OF THE DODECAGON INSCRIBE A
STAR AND A ROSETTE OF TWENTY-FOUR POINTS. WITHIN THE HEXAGON INSCRIBE
TWO TRIANGLES FROM A TANGENT CIRCUMFERENCE. A LOZENGE IS INSCRIBED
BETWEEN OPPOSITE SIDES OF THE SQUARE. LASTLY, BY _tricèles révolvés_,
ALL THE PROLONGED LINES ARE RECONCILED.]
[Illustration: 128 DISTRIBUTION OF THE DODECAGON, HEXAGON, AND SQUARE
ASSEMBLED. IN THE DODECAGON INSCRIBE A ROSETTE OF TWENTY-FOUR POINTS; IN
THE HEXAGON A ROSETTE OF TWELVE POINTS; AND, LASTLY, IN THE SQUARE A
ROSETTE OF EIGHT POINTS. A LITTLE PENTAGONAL STAR RECONCILES THE
PROLONGED LINES.]
[Illustration: 129 SUB-DIVIDE AS IN PLAN. DODECAGON, HEXAGON, AND SQUARE
ASSEMBLED. IN THE DODECAGON A ROSETTE OF TWELVE POINTS. IN THE TANGENT
CIRCUMFERENCE SIX HEXAGONS. THE SQUARE GOVERNS THE PROLONGED LINES.]
[Illustration: 130 THE ROSETTES ARE AS IN NUMBER 129. THE SMALLER
HEXAGONS GOVERN THE PRINCIPAL FIGURES.]
[Illustration: 131 SQUARE PLAN; DISTRIBUTION FOLLOWS THE NET-WORK OF THE
OCTAGON AND SQUARE. THE ROSETTE IS LINKED BY THE SMALLER OCTAGONS.]
[Illustration: 132 ANALOGOUS TO NUMBER 131.]
[Illustration: 133 OCTAGONS AND SQUARES ASSEMBLED. THE OCTAGONAL ROSETTE
GOVERNS.]
[Illustration: 134 SUB-DIVIDE AS INDICATED. DESCRIBE A CIRCLE IN WHICH
IS INSCRIBED A STAR WITH SIDES PROLONGED, DETERMINING AN OCTAGONAL
ROSETTE. BY CONCENTRIC CIRCLES, SOMEWHAT ARBITRARY, THE ROSETTE OF
SIXTEEN POINTS IS DETERMINED.]
[Illustration: 135 SUB-DIVIDE AS IN PLAN. IN A CIRCLE IS INSCRIBED AN
OCTAGONAL ROSETTE, AND TAKE A TANGENT CIRCUMFERENCE IN WHICH TO INSCRIBE
THE ROSETTE OF SIXTEEN POINTS. THE REST FOLLOWS.]
[Illustration: 136 DISTRIBUTION OF THE OCTAGON AND SQUARE. A CONCENTRIC
CIRCLE CONTAINS ARCS COMPOSING A CURVILINEAR ROSETTE, WITHIN WHICH IS A
ROSETTE OF FIVE POINTS. IN THE CENTRE OF THE SQUARE PLAN A CURVILINEAR
ROSETTE ENCLOSING AN OCTAGONAL ROSETTE WITH SIDES PROLONGED, BRINGING
INTO ACCORD THE PENTAGONAL ROSETTES.]
[Illustration: 137 SUB-DIVIDE THE MAIN CIRCLE INTO THIRTY-TWO EQUAL
PARTS, AND INSCRIBE THE ROSETTE OF SIXTEEN POINTS. OTHER CIRCLES ARE
INSCRIBED, EACH CONTAINING A PERFECT OCTAGON.]
[Illustration: 138 NET-WORK OF OCTAGON AND SQUARE. INSCRIBE A ROSETTE OF
EIGHT POINTS, AND IN A CIRCUMFERENCE TANGENT INSCRIBE A ROSETTE OF
TWENTY-FOUR POINTS. THE ROSETTES ARE BROUGHT INTO ACCORD BY THE
HEXAGONS.]
[Illustration: 139 IN A CIRCUMFERENCE INDICATED IN THE SQUARE PLAN
INSCRIBE THE OCTAGONAL ROSETTE. EXTENDED SIDES DETERMINE THE PERFECT
OCTAGONS.]
[Illustration: 140 SQUARE PLAN. OCTAGONAL AND SQUARE DISTRIBUTION. TWO
ROSETTES, ONE OF SIXTEEN AND ONE OF EIGHT POINTS. THE DIAGONALS FROM THE
ANGLES OF THE PENTAGON COMPLETE THE ROSETTE OF SIXTEEN POINTS, AND THE
EXTENDED SIDES OF THE PENTAGON DETERMINE THE OCTAGONAL ROSETTE.]
[Illustration: 141 SQUARE PLAN. NET-WORK OF THE OCTAGON AND SQUARE
ASSEMBLED. DESCRIBE CIRCUMFERENCES AS IN THE PLAN. IN JOINING THE POINTS
OF DIVISION, THERE IS ON THE ONE PART THE HEXAGON, AND ON THE OTHER PART
THE ROSETTE OF SIXTEEN POINTS. THE _tricèle révolvé_ ASSISTS IN
ELUCIDATING THE NET-WORK.]
[Illustration: 142 CURVILINEAR NET-WORK COMPOSED OF STARS OF FIVE, SIX,
AND EIGHT POINTS. DESCRIBE THE CIRCUMFERENCES AS IN THE PLAN. ONE CIRCLE
DESCRIBES THE STAR OF FIVE POINTS; ANOTHER CIRCLE INSCRIBES A
CURVILINEAR IRREGULAR HEXAGON; STILL ANOTHER CIRCLE INSCRIBES A STAR OF
EIGHT POINTS, AND THE LAST CIRCLE A STAR OF SIX POINTS.]
[Illustration: 143 SUB-DIVIDE AS INDICATED. THE PROLONGED LINES OF THE
PENTAGONAL STAR DETERMINE THE DECAGONAL STAR AND ROSETTE.]
[Illustration: 144 DIVIDE AS IN THE PLAN. BY TANGENT CIRCUMFERENCES
INSCRIBE THE ROSETTES. BY SIDES EXTENDED THE REST FOLLOWS.]
[Illustration: 145 OCTAGONAL, DECAGONAL, AND DODECAGONAL STARS AND
ROSETTES. BY SIDES PROLONGED THE REST FOLLOWS.]
[Illustration: 146 OCTAGONAL, DECAGONAL, AND DODECAGONAL STARS AND
ROSETTES. BY EXTENDED LINES AND THE ADJUSTMENT OF THE REGULAR OCTAGONS
THE DIAGRAM IS COMPLETED.]
[Illustration: 147 IN A CIRCUMFERENCE AS INDICATED INSCRIBE A STAR OF
NINE POINTS, AND BY SIDES PROLONGED A ROSETTE OF NINE POINTS; IN A
SECOND A STAR AND ROSETTE OF TWELVE POINTS; AND IN A THIRD CIRCUMFERENCE
A STAR AND ROSETTE OF TEN POINTS. THE REST FOLLOWS BY EXTENSION.]
[Illustration: 148 THE CENTRAL CIRCUMFERENCE DIVIDED INTO THIRTY-TWO
EQUAL PARTS PRODUCES A STAR AND ROSETTE OF SIXTEEN POINTS. DIVIDE OTHER
CIRCUMFERENCES TO PRODUCE STARS AND ROSETTES OF TWELVE AND TEN POINTS.
THE REST FOLLOWS.]
[Illustration: 149 CIRCUMFERENCES DIVIDED AS INDICATED. INSCRIBE STARS
OF NINE, TEN, AND TWELVE POINTS.]
[Illustration: 150 NET-WORK ANALOGOUS TO NUMBER 148. THE LINES OF THE
HEPTAGON, EXTENDED, JOIN THE LINES OF THE ROSETTES.]
[Illustration: 151 THIS EXAMPLE EXHIBITS THE MARVELLOUS INGENUITY OF THE
ARABIAN DESIGNER IN COMPOSITION, ROSETTES OF FIVE, SIX, SEVEN, AND EIGHT
POINTS BEING ADJUSTED. THE DESIGN IS EVIDENTLY THE PRODUCTION OF AN
ART-WORKMAN. IF THE NET-WORK IS NOT ACTUALLY PERFECT, IT APPROACHES
PERFECTION SO NEARLY THAT IT MAY BE CONSIDERED EXACT.]
[Illustration: 152 ISOCELES OR LOZENGE PLAN. THE ROSETTE OF FOURTEEN
POINTS RESULT FROM THE EXTENDED LINES OF THE HEPTAGONS.]
[Illustration: 153 ANALOGOUS TO NUMBER 152, BUT WITH DIAGONALS SOMEWHAT
DIFFERENTLY TREATED.]
[Illustration: 154 ROSETTE OF FOURTEEN POINTS GOVERNED BY HEPTAGONS.]
[Illustration: 155 ROSETTE OF FOURTEEN POINTS FROM EXTENDED LINES OF THE
HEPTAGONS. THE REST FOLLOWS.]
[Illustration: 156 PENTAGONAL ADJUSTMENT. A CIRCUMFERENCE TANGENT TO
THOSE OF THE PENTAGON INSCRIBES A STARRED ROSETTE OF FOURTEEN POINTS.]
[Illustration: 157 BY A CONCENTRIC CIRCUMFERENCE IS INSCRIBED THE
ROSETTE OF FOURTEEN POINTS. THE PENTAGON WHICH GOVERNS HAS ONE OF ITS
SIDES EXTENDED TO GREATER LENGTH THAN THE SIX OTHERS.]
[Illustration: 158 BY THE AID OF A CIRCUMFERENCE, INDICATED, IS
INSCRIBED THE HEPTAGONAL STAR FROM WHICH THE OTHER FIGURES PROCEED.]
[Illustration: 159 DECAGONAL STARS AND ROSETTES, WITH INTERCALARY
PENTAGONS. ALSO WITH INTERCALARY MESHES, WHICH ARE EQUAL TO THOSE OF THE
ROSETTES.]
[Illustration: 160 ANALOGOUS TO NUMBER 159.]
[Illustration: 161 SUB-DIVIDE THE SPACE SURROUNDING THE ISOCELES
TRIANGLE AS INDICATED. THE CIRCUMFERENCES BEING DESCRIBED. THE DIAGONALS
EXTENDED COMPLETE THE FIGURE.]
[Illustration: 162 PLAN AND CIRCUMFERENCES THE SAME AS NUMBER 161. THE
AID OF A CONCENTRIC CIRCUMFERENCE IS CALLED IN TO FORM THE ROSETTE OF
TEN POINTS. THE REST FOLLOWS.]
[Illustration: 163 DESCRIBE EQUAL AND TANGENT CIRCUMFERENCES TO FORM A
STAR OF TEN POINTS, AND TAKE THE DIAGONALS OF SIX IN SIX DIVISIONS.]
[Illustration: 164 DESCRIBE CIRCUMFERENCES AS INDICATED, AND TAKE THE
DIAGONALS OF SIX IN SIX DIVISIONS. DRAW A HORIZONTAL LINE AT THE HIGHER
LINE OF THE LITTLE PENTAGON, AND REPEAT THE CONSTRUCTION BELOW THE
LINE.]
[Illustration: 165 DIVIDE CIRCUMFERENCES AND DRAW PARALLEL LINES AS
INDICATED. TAKE THE DIAGONALS OF SIX IN SIX DIVISIONS, AND THE REST
FOLLOWS.]
[Illustration: 166 DESCRIBE CIRCUMFERENCES EQUAL AND TANGENT, AND TAKE
THE DIAGONALS OF FOUR IN FOUR DIVISIONS; THEN IN THE CONCENTRIC
CIRCUMFERENCES TAKE THE DIAGONALS OF SIX IN SIX DIVISIONS.]
[Illustration: 167 SUB-DIVIDE AS IN PLAN. INSCRIBE THE ROSETTE OF TEN
POINTS; THE LITTLE PENTAGONS AND LOZENGES FOLLOW, AND THE REST FOLLOWS.]
[Illustration: 168 ANALOGOUS TO NUMBER 167, BUT IN PLACE OF THE ROSETTE
STARS ARE FORMED.]
[Illustration: 169 CURVILINEAR TRANSFORMATION OF NUMBERS 167 AND 168.
THE POINTS OF CENTRAL DISTRIBUTION ARE MARKED BY MINUTE CROSSES.]
[Illustration: 170 SIMILAR GROUND WORK TO THE THREE FOREGOING DIAGRAMS.
THE LITTLE PENTAGONS GOVERN THE DESIGN.]
[Illustration: 171 DESCRIBE THE CIRCUMFERENCES AS IN PLAN, AND INSCRIBE
THE DECAGONAL STAR. THE SMALL DECAGONS IN THE CENTRE OF THE TRIANGLES OF
THE PLAN, BY EXTENDED LINES, FORM THE ROSETTE.]
[Illustration: 172 DIVIDE AS IN PLAN. UPON THE LESSER SIDE TRACE AN
ISOCELES TRIANGLE. BY THE APEX OF THE TRIANGLE TRACE A CIRCUMFERENCE, IN
WHICH TAKE THE DIAGONALS OF THREE IN THREE DIVISIONS.]
[Illustration: 173 LOZENGE PLAN, WITH COMBINING LINES. THE LOZENGE IS
IMAGINARY, AND DOES NOT INDICATE THE RADII IN EXTENSION ONE WITH THE
OTHER, BUT ONLY THE ORDER OF SUB-DIVISION OF THE PENTAGONAL AND
DECAGONAL ROSETTES.]
[Illustration: 174 SUB-DIVIDE AS IN PLAN THE SPACE SURROUNDING A POINT
INTO TWENTY EQUAL PARTS. AT THE MEETING OF THE HORIZONTAL WITH THE
VERTICAL LINE DRAW A CONCENTRIC CIRCUMFERENCE IN WHICH IS INSCRIBED THE
STARRED DECAGONAL ROSETTE. THE SMALL PENTAGON IS DOMINANT.]
[Illustration: 174 SAME GROUND-WORK AS NUMBER 172, BUT WITH A GREATER
INTERVAL BETWEEN THE ROSETTES. ONE OF THE ROSETTES, INSTEAD OF BEING
RECTILINEAR, IS CURVILINEAR.]
[Illustration: 175 SUB-DIVIDE AS INDICATED. DESCRIBE CIRCUMFERENCES
EQUAL AND TANGENT, AND TAKE THE DIAGONALS OF FOUR IN FOUR DIVISIONS.
LASTLY, THE ROSETTES ARE EFFECTED.]