Suchergebnisse

Denne Ph.D. afhandling omfatter fundamentale betragtninger omkring topologier, algoritmer, implementeringer, metoder etc., der kan indg˚a i næste generation af aktive kontrol systemer. Specifikt foresl˚as der en variant af feedforward kontrol refereret til som indesluttet feedforward aktiv kontrol forkortet IFFAK. I denne topologi indg˚ar et sæt reference sensorer, der er positioneret p˚a en overflade, der fuldt ud indeslutter de ønskede stille-zoner, hvori et sæt performance sensorer monitorerer den opn˚aede støjreduktion. Denne indesluttet-feedforward aktiv kontrol (IFFAK) topologi er indlejret i et mange-input-mange-output (MIMO) system, der omfatter b˚ade feedforward og feedback kontrol. Det totale system er refereret til som et hybrid MIMO indesluttetfeedforward FBS (HMIMOIFFFBS). Undersøgelsen af et komplekst multi-kanals aktiv støjreduktion (ASR) system med hybrid feedforward og feedback topologier er motiveret ud fra krav om høj aktiv støjdæmpning i ekstreme støjmiljøer, som f.eks. opleves ombord p˚a luftb˚arne militære platforme. Støjoptagelser erhvervet ombord p˚a s˚adanne fartøjer afslører lydtryk, der ofte overstiger 140 dB re. 20 μPa. Endvidere udviser disse støjsignaler store tidslige s˚avel som spatiale variationer. Naturlige begrænsninger i feedback baserede aktiv kontrol (AK) systemer som typisk anvendes i moderne ASR støjværn, hvor modstøjssignalet notorisk er forsinket i forhold til den primære forstyrrelse, sætter en øvre grænse for, hvor stor en aktiv dæmpning, der kan opn˚aes. S˚aledes, hersker der i krævende militære applikationer et krav om nye mere avancerede og effektive ASR løsninger. Den opn˚aelige ASR i et FFS er i stor udstrækning bestemt af kohærensen mellem sættet af reference sensorer og sættet af fejl- eller performance sensorer. S˚aledes omfatter denne afhandling en del kohærensundersøgelser baseret p˚a diffustfeltsm˚alinger i et støjkammer samt m˚alinger, der er foretaget i en CH-47D Chinook helikopter. Fra disse kohærensanalyser kan det konkluderes, at IFFAK systemet anvendt p˚a pilothjælme giver mulighed for ca. 25 dB støjreduktion ved 100 Hz faldende til ca. 10 dB dæmpning ved 900 Hz. Endvidere, er der ikke nogen umiddelbare tegn p˚a en mætning med stigende antal reference sensorer. S˚aledes vil et større antal reference sensor forventeligt kunne øge den øvre ASR frekvensgrænse for systemet, der bestemmes af den rumlige samplingstæthed. I hybridsystemet indg˚ar der b˚ade et kontinuerlig-tids FBS og et diskret-tids FBS. Disse vil bidrage med yderligere støjreduktion primært overfor bredb˚andet støj henholdsvis overfor periodiske signaler. Tidsforsinkelser udgør en anden bestemmende faktor for den opn˚aelige effekt i et FFS design, men specielt i et FBS design, eftersom fysiske systemer altid opererer kausalt. For at vurdere størrelsesordenen af det tidsforspring som hver reference sensor giver i forhold til hver fejlsensor i det foresl˚aede IFFAK system indføres en størrelse, der betegnes som den spatialt-vægtet middeltidsgevinst. Der eksisterer imidlertid et problem, n˚ar man forsøger at modellere et fysisk system med en endelig rummelig udstrækning og hvor der s˚aledes ikke er nogen indlysende input-output definition med et endeligt-element multi-kanals system. Som regel eksisterer der ikke nogen unik overføringsfunktion eftersom systemet ikke bliver punktvist stimuleret, men derimod stimuleret xii over et areal som for eksempel under diffustfelts belysning. En ny akustisk signalbehandlingsmetode, der betegnes som samlet kanal residual spektral analyse (SKRSA) er udviklet. Denne metode benyttes til ekstraktion af fælles signal information fra forskellige observationspunkter i rummet. Ideen er at separere hvert spektrum i et kohærent spektrum og et residual spektrum. Indholdet i det kohærente spektrum kan opn˚aes som en linear kombination af spektrene fra de andre kanaler, hvorimod indholdet af det residuale spektrum er unikt for den p˚agældende kanal. I et specifikt eksempel, belyses et system best˚aende af et sæt reference sensorer monteret p˚a en Gentex HGU-55/P hjælm, der igen er p˚amonteret en hoved og torso simulator med et diffust lydfelt. Under anvendelse af SKRSA metoden estimeres den spatialt-vægtet middeltidsgevinst til at være i størrelsesorden 800-900μs. Afhandlingen omfatter ogs˚a en detaljeret beskrivelse af en ny id´e til en beregningsmæssig effektiv implementering af et multi-kanals system, hvor b˚ade de adaptive filtre, der indg˚ar i den aktive kontrol s˚avel som de adaptive filtre, der indg˚ar til modellering af systemoverføringsfunktionerne kan antage individuelle længder. En ny og mere generel variant af APA algoritmen er udviklet. Denne adaptive filter algoritme inkluderer parametre for b˚ade wægt-styret og kontrol-effekt-styret lækage, adaptiv tap-vægte regulering s˚avel som numerisk regulering og betegnes MC-αγΠ-APA. En simplificering af denne algoritme, fører til MC-αγΠ-NLMS algoritmen, der er en udbygget variant af NLMS algoritmen. Systemets evne til off-line simultant at kunne identificere et complex system best˚aende af ialt 28 enkelt systemgrene bliver demonstreret. Forskellig adaptive filtre samt parametering heraf bliver udforsket. Et nyt og generelt multi-hastigheds systemkoncept for aktiv kontrol er udviklet. Specifikt implementeres og testes et system, hvor der i alt samples med tre forskellige hastigheder. P˚a multi-hastighedsniveau 0 benyttes en meget høj samplingsfrekvens med henblik p˚a at reducere forsinkelser i konverteringstrinene, der indg˚ar i de sekundære grene. Den ikke adaptive kontrol udføres p˚a det lavere multi-hastigheds niveau 1. Herved tilsikres et kompromis imellem forsinkelser til afgivelse af modstøjssignaler og krav til en endelig system b˚andbredde. Sluttelig foreg˚ar den adaptive kontrol ved det lavere multi-hastigheds niveau 2. Herved begrænses den ofte beregningsmæssige tunge adaptive filter opdatering til en s˚a lav samplingsfrekvens som muligt. I et specifikt eksempel demonstreres, at en beregningsmæssig besparelse p˚a ca. 40% kan opn˚as under opretholdelse af samme ASR ved nedsampling fra multi-hastighedsniveau 1 p˚a 24 kHz til multi-hastighedsniveau 2 p˚a 3 kHz. Det er en almindelig ingeniørpraksis at foretage en antagelse om Gaussisk fordelte signaler. Imidlertid, er mange fænomener i dagligdagen bedst modelleret med s˚akaldte alfa-stabile fordelings funktioner. Dette gælder ogs˚a for støjsignaler, der ønskes undertrykt ved hjælp af et aktivt støjdæmpningssystem. Afhandlingen indholder en kort teknisk beskrivelse af de stabile fordelingsfunktioner samt adaptive filter algoritmer for disse type signaler. Store dele af HMIMOIFFFB systemet samt de udviklede metoder og algoritmer er implementeret i et realtids miljø, der inkluderer en signal processor. I første omgang vil disse blive aftestet p˚a en til form˚alet designet aktive kontrol testenhed. ; This Ph.D. thesis includes fundamental considerations about topologies, algorithms, implementations, methods etc., that can enter in the next generation of active control (AC) systems. Specifically, a new variant of feedforward control referred to as confined feedforward active control (CFFAC) is proposed. This topology is constituted from a set of reference sensors that are positioned on a surface that completely confines the desired zones of quite. A set of performance sensors monitors the achieved noise reduction. This CFFAC topology in turn is embedded in a multiple-input and multiple-output (MIMO) system that facilitates both feedforward and feedback control. The general system is then referred to as hybrid MIMO confined-feedforward feedback (HMIMOCFFFB) active noise reduction (ANR) system. The investigation of a multi-channel ANR system with hybrid feedforward and feedback topologies is motivated by requirements of high ANR attenuation in extreme noise environments as typically experienced onboard airborne military platforms. Noise recordings acquired on such platforms reveal very high sound pressure levels often exceeding 140 dB re. 20 μPa. Moreover, these noise signals exhibit large temporal as well as spatial variations. Inherent limitations are related to the use of stand-alone feedback AC implementation commonly applied in modern ANR headset. In such systems the anti-noise signal is notoriously behind the primary disturbance in time. Accordingly, in demanding military applications requirements on more advanced and effective ANR system designs prevail. The achievable ANR performance in a feedforward system (FFS) is to a large extent determined by the degree of coherence between the set of reference sensors and the set of error sensors (or performance sensors). Accordingly, this thesis includes a number of coherence analysis that are based on diffuse sound field measurements in a reverberant chamber and measurements conducted onboard a CH-47D Chinook helicopter. From these coherence analysis it can be concluded that the CFFAC system with 10 reference sensors applied to pilot helmets potentially provides approximately 25 dB noise reduction at 100 Hz decreasing to approximately 10 dB attenuation at 900 Hz. Moreover, there is no apparent sign of saturation of the noise reduction with an increasing number of reference sensors. Accordingly, by using more reference sensors the spatial sampling rate is increased which in turn most likely also will lead to an increased ANR bandwidth. The hybrid system is also constituted from a continuous-time feedback system (FBS) and a discrete-time FBS. The continuous-time FBS is primarily responsible for additional broadband noise reduction, whereas the discrete-time FBS primarily is responsible for the attenuation of periodic signals. Owing to the requirement on causal operation of a physical AC system time delays will also to a large extent determine the achievable performance in FFS design and in particular in FBS design. A quantity referred to as the spatially-weighted-averaged acquisition lead time is introduced to represent the averaged time-advance obtained by each reference sensor relative to each performance sensor involved in the proposed CFFAC system. A problem exist when one attempts to model a physical spatially distributed system with no obvious input and output channel definition by a finite lumped-elements multi-channel system. Usually, no unique transfer function x exist as the system is not point-wise excited, but excited over an area as in the case of diffuse sound field illumination. A new method for acoustical signal processing that is referred to as joint-channel residual spectral analysis (JCRSA) is developed. The JCRSA method is used for the extraction of joint signal information from different observation positions in space. The idea is to separate each spectrum in a coherent spectrum and a residual spectrum. The contents of the coherent spectrum can be obtained from a linear superposition of the other signals, whereas the residual spectrum bears information that is unique to each specific channel. In a specific example a system consisting of 10 reference sensors flush-mounted on a Gentex HGU-55/P helmet that in turn is mounted on a head and torso simulator (HATS), is exposed to diffuse sound field illumination. By applying the JCRSA method the spatially-weighted-averaged acquisition lead times provided by the reference sensors relative to the performance sensors are estimated to be as much as 800-900μs. The thesis also includes a detailed description of a new idea for a computational efficient implementation of a multi-channel system in which the adaptive filters for adaptive control as well as the adaptive filters used for plant modeling are allowing to take different lengths. A new and more general variant of the affine projection algorithm has been developed. This adaptive filter algorithm that is denoted by multiple-channel-αγΠ-affine projection algorithm includes parameters for both weight-driven and control-effort-driven leakage, adaptive tap-weight regularization as well as numerical regularization. A simplification of this algorithm leads to the MC-αγΠ-NLMS algorithm that is an extended variant of the NLMS algorithm. Off-line simultaneous system identification capabilities of a complex system involving a total 4 secondary paths, 20 feedback paths and 4 control-performance paths is demonstrated. Different adaptive filters and parameterizations hereof are examined. A novel and general multi-rate adaptive filter for adaptive AC has been developed. Specifically, a system involving 3 different sampling rates has been implemented and the results hereof are presented. In this multi-rate system conversion take place at highly oversampled rates in order to reduce the delays in the secondary paths. The non-adaptive control is performed at a somewhat lower rate. Hereby, a compromise between delays related to the generation of the anti-noise signal and the computational load involved is ensured. Finally, the adaptive control that might be computational intensive takes place at an even slower sampling rate hereby relaxing the requirements on a high bandwidth. It is demonstrated that computational savings as high as 40% can be achieved in a 192, 24, 3 kHz triple-rate system as compared with a 24 kHz single-rate system without sacrificing the ANR performance. It is common engineering practice to apply an assumption of Gaussian distributed signals. However, many phenomena encountered in daily life fall into a generalization of the normal distribution that is referred to as α-stable distributions. Noise sources encountered in the domain of AC are sometimes best fitted to the family of α-stable distributions. This thesis includes a brief technical introduction to the stable distributions and description of the adaptive filter that can be used for AC. Large parts of the HMIMOCFFFB system including the developed methods and algorithms have been implemented in a real-time environment (RTE) that includes a signal processor. Test on the helmet system will continue and a dedicated reference test unit (RTU) for AC is currently being designed.

El aumento progresivo de la población, la industrialización y el consumismo son los factores principales por los que ha incrementado la generación de residuos en los últimos años, requiriendo de una gestión integral para proteger la salud pública y el medio ambiente. La gestión y tratamiento adecuado de los residuos sólidos urbanos constituye, actualmente, un problema clave en materia de sostenibilidad, ya que es necesario dar solución a un problema ambiental, económico, social y sanitario, evitando la clásica deposición en vertederos. Sin embargo, el tratamiento de residuos orgánicos incrementa la contaminación por olores desagradables. Dichas emisiones han de ser evaluadas, cualitativa y cuantitativamente, con el propósito de minimizarlas. Actualmente se utilizan diversas herramientas y dispositivos de muestreo que permiten, según las normativas aplicables en el seguimiento de olores (EN 13725 y VDI 3880), llevar a cabo la evaluación de los procesos que generan malos olores y cuantificar las emisiones para dar una magnitud real de cuánto olor emite un foco emisor. Para analizar los compuestos olorosos, en focos emisores, la olfatometría dinámica se ha establecido como una técnica sensorial adecuada que cuantifica la concentración de olor de una muestra olorosa (ouE/m3). Sin embargo, los materiales utilizados en el almacenamiento de las muestras gaseosas pueden no ser estancos para ciertos compuestos olorosos. En este sentido, y en colaboración con el Laboratory of Industrial Environment Engineering (Ecole des Mines d'Alès, Francia), se ha realizado un estudio de los efectos de permeabilidad y adsorción de diversos compuestos de azufre volátiles a través de las bolsas de muestreo de Nalophan®. El sulfuro de hidrógeno (H2S), metilmercaptano (metil-SH) y disulfuro de carbono (CS2) son los que presentan una menor retención en el interior de las bolsas debido a su tamaño y estructura molecular, lo cual ha derivado en una pérdida global del 10% a las 30 h de almacenamiento y del 25% a las 95 h. Además, las válvulas necesarias para la recolección de muestras han dado lugar a pequeñas pérdidas de compuestos por adsorción. Con objeto de profundizar en las emisiones de olor generadas durante el tratamiento de residuos orgánicos, se han llevado a cabo estudios a escala de laboratorio, piloto e industrial de distintas materias residuales. A escala de laboratorio se ha realizado el seguimiento del proceso de co-compostaje en vasos Dewar de lodo de depuradora de aguas residuales municipales con distintas proporciones de residuo del cultivo de la planta de berenjena, insertando como novedad unas conducciones verticales de PVC perforadas para favorecer las condiciones aerobias del proceso. El pico de olor generado durante la etapa hidrolítica del proceso de compostaje se ha minimizado con una mayor proporción de residuos de berenjena en la mezcla con los lodos. Asimismo, la estabilidad del compost obtenido al final del proceso, demuestra la viabilidad de la tecnología para valorizar ambas materias residuales, al igual que la concentración de fósforo arroja mejores resultados, en cuanto a calidad del producto final, que el lodo compostado individualmente. Se ha demostrado que la reducción de la concentración de sólidos volátiles ha sido la principal causa en la emisión de compuestos olorosos durante el proceso, de forma más significativa que la eliminación de compuestos nitrogenados. A escala piloto se ha evaluado el proceso de compostaje en un respirómetro dinámico y, dado que las condiciones operacionales están más controladas que en una instalación industrial, se han realizado estudios con diversas materias residuales (lodo de depuradora, cáscara de naranja, residuo de la manufactura del pescado, residuo del extrusionado de fresa y fracción orgánica de residuos sólidos urbanos) evaluando, mediante la aplicación de herramientas estadísticas avanzadas, la importancia de las variables del proceso sobre el impacto oloroso. En este sentido, un análisis de componentes principales de la matriz de datos obtenida ha permitido clasificar los sustratos por su origen, siendo el índice respirométrico dinámico (proporcional a la velocidad de consumo de oxígeno en la biodegradación) y la tasa de emisión de olor las variables más influyentes. Otra de las herramientas estadísticas aplicada es la regresión multivariante, demostrando ser una técnica adecuada en la predicción de concentración y/o tasa de emisión de olores. La regresión multivariante ha sido una herramienta a partir de la cual se han evaluado las variables operacionales más influyentes en la generación de olores. Además, se han determinado los grupos funcionales, asociados a los compuestos presentes en las materias residuales mediante la aplicación de la novedosa tecnología NIR, encontrándose una clara relación entre estas y las emisiones de olor generadas durante el tratamiento. A escala industrial, se ha elaborado un mapa global de olor de una planta de tratamiento de residuos sólidos urbanos de Córdoba capital, en el que se han identificado los puntos críticos de emisión de olor y las variables más influyentes en su generación. En este sentido, la recepción de basura orgánica y de lodos de depuradora (procedentes de la planta de tratamiento de aguas residuales "La Golondrina") son los puntos críticos más olorosos, con tasas de emisión de 14,57 y 2,41 ouE/s·m2, respectivamente. Además, considerando la variabilidad de puntos críticos en la planta, y tras realizar un estudio global de ellas, las variables respirométricas, la concentración de nitrógeno y el tiempo de residencia de los materiales residuales han destacado como las más influyentes en el impacto oloroso. Finalmente, también se ha estimado dicho impacto oloroso en una planta de compostaje que gestiona lodos de depuradora (pretratado y sin pretratar) y residuos de mercado mediante compostaje en pilas, en términos de concentración de inmisión, con el propósito de evaluar la generación y posterior dispersión de las emisiones olorosas en las zonas colindantes. A partir de un modelo de dispersión Gaussiano de pluma se han obtenido los perfiles de inmisión en función de la distancia de la planta considerando la orografía, condiciones meteorológicas y atmosféricas más desfavorables. La concentración de inmisión máxima observada no ha superado en ningún caso los límites establecidos en el borrador del Anteproyecto de ley contra la contaminación odorífera de Cataluña de 3 ou/m3, con una estabilidad atmosférica neutra y una velocidad del viento de 2,6 m/s. Además, el seguimiento del proceso de compostaje a partir de las variables tradicionales ha demostrado que las emisiones olorosas derivadas de la planta de tratamiento y gestión de residuos sólidos orgánicos mediante compostaje están nuevamente relacionadas con la volatilización de nitrógeno y la eliminación de los sólidos volátiles. Los novedosos resultados obtenidos en los trabajos de investigación que componen esta Tesis Doctoral suponen un avance científico significativo en términos de cuantificación y determinación de emisiones olorosas derivadas del tratamiento de residuos. La aplicación de las nuevas herramientas utilizadas en este trabajo podrían contribuir a mitigar el impacto oloroso que genera la gestión y tratamiento de residuos, un proceso que a día de hoy se considera indispensable y que trata de dar una nueva vida a los millones de toneladas de materia orgánica que cada día genera el ser humano. ; The progressive increase in population, industrialization and consumerism are the main factors leading to the enhancement of waste generation during the last years, which requires integrated management to protect public health and environment. The adequate management and treatment of urban solid waste is nowadays a major issue in terms of sustainability, leading to the necessity of solving an environmental, economic, social and sanitary problem by avoiding classical landfilling. Nevertheless, the treatment of organic waste promotes pollution by the generation of unpleasant odor. The emission of odor should be evaluated qualitative and quantitatively with the aim of achieving its minimization. Several sampling tools and devices are currently being used in accordance with the legislation on odor monitoring (EN 13725 and VDI 3880) to evaluate those processes that generate unpleasant odor and to quantify their emission by providing a real magnitude on how much odor an emission source generates. Dynamic olfactometry has been stablished as an adequate sensorial technique to analyze the odor concentration in an odorous sample (ouE/m3). However, the materials used for storing gaseous samples might not be watertight for certain odorous compounds. In this context, and in collaboration with the Laboratory of Industrial Environment Engineering (Ecole des Mines d'Alès, France), a study of the permeability and sorption effects of several volatile sulfur compounds stored in sampling Nalophan® bags was carried out. Hydrogen sulfide (H2S), methyl mercaptan (methyl- SH) and carbon disulfide (CS2) showed the lowest retention inside the sampling bags due to their molecular size and structure, which led to global loss of 10% after 30 h of storage time and 25% at 95 h. Furthermore, the valves necessary to store gaseous samples led to small loss due to sorption processes. With the aim of deepening in odorous emissions derived from the treatment of organic waste, several research studies of different residual substrates were carried out at laboratory, pilot plant and industrial scales. The monitoring process of co-composting sewage sludge from wastewater treatment at different mixing proportions with eggplant waste was carried out in Dewar vessels at laboratory scale. As a novel aspect, vertical and perforated PVC pipes were inserted in the vessels to favor aerobic conditions inside the substrates being composted. The odor peak generated during the hydrolytic stage of composting was minimized by increasing the proportion of eggplant waste in the mixture with sewage sludge. Likewise, the stability of the final product demonstrates the feasibility of cocomposting to valorize both residual substrates simultaneously. Furthermore, the concentration of phosphorus was higher in the final compost than the value obtained when sewage sludge was composted individually. The reduction of the concentration of volatile solids was found to be the main cause of odor emission, being more influential than the removal of nitrogenous compounds. On the other hand, the composting process was also evaluated in a dynamic respirometer at pilot plant scale. As the operational conditions were more controlled than at industrial scale, several studies were carried out with several residual substrates (sewage sludge, orange peel, fish waste, strawberry extrudate and organic fraction of municipal solid waste) to evaluate the importance of the process variables on the odorous impact through advanced statistical tools. In this sense, a principal components analysis applied to the data matrix obtained allowed classifying residual substrates by origin, with the dynamic respirometric index (proportional to the oxygen consumption rate during biodegradation) and the odor emission rate being the most influential variables. Another statistical tool used was multivariate regression, which has been demonstrated to be an adequate technique to predict odor concentration and/or emission rate. Multivariate regression was used to evaluate the most influential operational variables in odor generation. In addition, the functional groups associated to the compounds contained in the residual substrates were determined by the innovative NIR technology. A relationship between functional groups and odor emission generated during waste treatment was identified. At industrial scale, a global odor map of a solid waste plant that treats residues generated in the city of Cordoba was elaborated. Such a map identified the critical odor emission points and the most influential variables in its generation. Reception of organic waste and sewage sludge derived from the wastewater treatment plant "La Golondrina" were found to be the most critical points in terms of odor generation, with odor emission rates up to 14.57 and 2.41 ouE/s·m2, respectively. Furthermore, after considering the variability of critical points inside the plant and having carried out a global analysis, respirometric variables, nitrogen concentration and residence time of residual substrates were identified as the most influential variables on the odorous impact. Finally, the odorous impact derived from a plant that manages sewage sludge (raw and pretreated) and market waste through composting in piles was estimated in terms of immision concentration, with the aim of evaluating the generation and subsequent dispersion of odorous emissions in neighboring areas. By applying a plume Gaussian dispersion model, the immision profiles as a function of distance were obtained considering orography and the most unfavorable weather and atmospheric conditions. The maximal immision concentration did not exceed the threshold stablished by the draft law against odorous pollution in Cataluña (3 ou/m3), at neutral atmospheric stability and wind speed of 2.6 m/s. In addition, monitoring composing process through traditional variables demonstrated again that odorous emissions derived from the treatment and management of organic solid waste are related to nitrogen volatilization and volatile solids removal. The innovative results reported in the research studies included in this Doctoral Thesis have led to an important scientific advance in terms of quantification and determination of odorous emissions derived from waste treatment. The application of new tools developed in this Thesis might contribute to mitigate the odorous impact derived from waste management and treatment, which is considered essential in current societies and tries to take the advantage of millions of tons of organic waste generated by human beings every day.

Publisher's version (útgefin grein) ; Background In an era of shifting global agendas and expanded emphasis on non-communicable diseases and injuries along with communicable diseases, sound evidence on trends by cause at the national level is essential. The Global Burden of Diseases, Injuries, and Risk Factors Study (GBD) provides a systematic scientific assessment of published, publicly available, and contributed data on incidence, prevalence, and mortality for a mutually exclusive and collectively exhaustive list of diseases and injuries. Methods GBD estimates incidence, prevalence, mortality, years of life lost (YLLs), years lived with disability (YLDs), and disability-adjusted life-years (DALYs) due to 369 diseases and injuries, for two sexes, and for 204 countries and territories. Input data were extracted from censuses, household surveys, civil registration and vital statistics, disease registries, health service use, air pollution monitors, satellite imaging, disease notifications, and other sources. Cause-specific death rates and cause fractions were calculated using the Cause of Death Ensemble model and spatiotemporal Gaussian process regression. Cause-specific deaths were adjusted to match the total all-cause deaths calculated as part of the GBD population, fertility, and mortality estimates. Deaths were multiplied by standard life expectancy at each age to calculate YLLs. A Bayesian meta-regression modelling tool, DisMod-MR 2.1, was used to ensure consistency between incidence, prevalence, remission, excess mortality, and cause-specific mortality for most causes. Prevalence estimates were multiplied by disability weights for mutually exclusive sequelae of diseases and injuries to calculate YLDs. We considered results in the context of the Socio-demographic Index (SDI), a composite indicator of income per capita, years of schooling, and fertility rate in females younger than 25 years. Uncertainty intervals (UIs) were generated for every metric using the 25th and 975th ordered 1000 draw values of the posterior distribution. Findings Global health has steadily improved over the past 30 years as measured by age-standardised DALY rates. After taking into account population growth and ageing, the absolute number of DALYs has remained stable. Since 2010, the pace of decline in global age-standardised DALY rates has accelerated in age groups younger than 50 years compared with the 1990-2010 time period, with the greatest annualised rate of decline occurring in the 0-9-year age group. Six infectious diseases were among the top ten causes of DALYs in children younger than 10 years in 2019: lower respiratory infections (ranked second), diarrhoeal diseases (third), malaria (fifth), meningitis (sixth), whooping cough (ninth), and sexually transmitted infections (which, in this age group, is fully accounted for by congenital syphilis; ranked tenth). In adolescents aged 10-24 years, three injury causes were among the top causes of DALYs: road injuries (ranked first), self-harm (third), and interpersonal violence (fifth). Five of the causes that were in the top ten for ages 10-24 years were also in the top ten in the 25-49-year age group: road injuries (ranked first), HIV/AIDS (second), low back pain (fourth), headache disorders (fifth), and depressive disorders (sixth). In 2019, ischaemic heart disease and stroke were the top-ranked causes of DALYs in both the 50-74-year and 75-years-and-older age groups. Since 1990, there has been a marked shift towards a greater proportion of burden due to YLDs from non-communicable diseases and injuries. In 2019, there were 11 countries where non-communicable disease and injury YLDs constituted more than half of all disease burden. Decreases in age-standardised DALY rates have accelerated over the past decade in countries at the lower end of the SDI range, while improvements have started to stagnate or even reverse in countries with higher SDI. Interpretation As disability becomes an increasingly large component of disease burden and a larger component of health expenditure, greater research and development investment is needed to identify new, more effective intervention strategies. With a rapidly ageing global population, the demands on health services to deal with disabling outcomes, which increase with age, will require policy makers to anticipate these changes. The mix of universal and more geographically specific influences on health reinforces the need for regular reporting on population health in detail and by underlying cause to help decision makers to identify success stories of disease control to emulate, as well as opportunities to improve. Copyright (C) 2020 The Author(s). Published by Elsevier Ltd. ; Research reported in this publication was supported by the Bill & Melinda Gates Foundation; the University of Melbourne; Queensland Department of Health, Australia; the National Health and Medical Research Council, Australia; Public Health England; the Norwegian Institute of Public Health; St Jude Children's Research Hospital; the Cardiovascular Medical Research and Education Fund; the National Institute on Ageing of the National Institutes of Health (award P30AG047845); and the National Institute of Mental Health of the National Institutes of Health (award R01MH110163). The content is solely the responsibility of the authors and does not necessarily represent the official views of the funders. The authors alone are responsible for the views expressed in this Article and they do not necessarily represent the views, decisions, or policies of the institutions with which they are affiliated, the National Health Service (NHS), the National Institute for Health Research (NIHR), the UK Department of Health and Social Care, or Public Health England; the United States Agency for International Development (USAID), the US Government, or MEASURE Evaluation; or the European Centre for Disease Prevention and Control (ECDC). This research used data from the Chile National Health Survey 2003, 2009-10, and 2016-17. The authors are grateful to the Ministry of Health, the survey copyright owner, for allowing them to have the database. All results of the study are those of the authors and in no way committed to the Ministry. The Costa Rican Longevity and Healthy Aging Study project is a longitudinal study by the University of Costa Rica's Centro Centroamericano de Poblacion and Instituto de Investigaciones en Salud, in collaboration with the University of California at Berkeley. The original pre-1945 cohort was funded by the Wellcome Trust (grant 072406), and the 1945-55 Retirement Cohort was funded by the US National Institute on Aging (grant R01AG031716). The principal investigators are Luis Rosero-Bixby and William H Dow and co-principal investigators are Xinia Fernandez and Gilbert Brenes. The accuracy of the authors' statistical analysis and the findings they report are not the responsibility of ECDC. ECDC is not responsible for conclusions or opinions drawn from the data provided. ECDC is not responsible for the correctness of the data and for data management, data merging and data collation after provision of the data. ECDC shall not be held liable for improper or incorrect use of the data. The Health Behaviour in School-Aged Children (HBSC) study is an international study carried out in collaboration with WHO/EURO. The international coordinator of the 1997-98, 2001-02, 2005-06, and 2009-10 surveys was Candace Currie and the databank manager for the 1997-98 survey was Bente Wold, whereas for the following surveys Oddrun Samdal was the databank manager. A list of principal investigators in each country can be found on the HBSC website. Data used in this paper come from the 2009-10 Ghana Socioeconomic Panel Study Survey, which is a nationally representative survey of more than 5000 households in Ghana. The survey is a joint effort undertaken by the Institute of Statistical, Social and Economic Research (ISSER) at the University of Ghana and the Economic Growth Centre (EGC) at Yale University. It was funded by EGC. ISSER and the EGC are not responsible for the estimations reported by the analysts. The Palestinian Central Bureau of Statistics granted the researchers access to relevant data in accordance with license number SLN2014-3-170, after subjecting data to processing aiming to preserve the confidentiality of individual data in accordance with the General Statistics Law, 2000. The researchers are solely responsible for the conclusions and inferences drawn upon available data. Data for this research was provided by MEASURE Evaluation, funded by USAID. The authors thank the Russia Longitudinal Monitoring Survey, conducted by the National Research University Higher School of Economics and ZAO Demoscope together with Carolina Population Center, University of North Carolina at Chapel Hill and the Institute of Sociology, Russia Academy of Sciences for making data available. This paper uses data from the Bhutan 2014 STEPS survey, implemented by the Ministry of Health with the support of WHO; the Kuwait 2006 and 2014 STEPS surveys, implemented by the Ministry of Health with the support of WHO; the Libya 2009 STEPS survey, implemented by the Secretariat of Health and Environment with the support of WHO; the Malawi 2009 STEPS survey, implemented by Ministry of Health with the support of WHO; and the Moldova 2013 STEPS survey, implemented by the Ministry of Health, the National Bureau of Statistics, and the National Center of Public Health with the support of WHO. This paper uses data from Survey of Health, Ageing and Retirement in Europe (SHARE) Waves 1 (DOI:10.6103/SHARE. w1.700), 2 (10.6103/SHARE.w2.700), 3 (10.6103/SHARE.w3.700), 4 (10.6103/SHARE.w4.700), 5 (10.6103/SHARE.w5.700), 6 (10.6103/SHARE.w6.700), and 7 (10.6103/SHARE.w7.700); see Borsch-Supan and colleagues (2013) for methodological details. The SHARE data collection has been funded by the European Commission through FP5 (QLK6-CT-2001-00360), FP6 (SHARE-I3: RII-CT-2006-062193, COMPARE: CIT5-CT-2005-028857, SHARELIFE: CIT4-CT-2006-028812), FP7 (SHARE-PREP: GA N degrees 211909, SHARE-LEAP: GA N degrees 227822, SHARE M4: GA N degrees 261982) and Horizon 2020 (SHARE-DEV3: GA N degrees 676536, SERISS: GA N degrees 654221) and by DG Employment, Social Affairs & Inclusion. Additional funding from the German Ministry of Education and Research, the Max Planck Society for the Advancement of Science, the US National Institute on Aging (U01_AG09740-13S2, P01_AG005842, P01_AG08291, P30_AG12815, R21_AG025169, Y1-AG-4553-01, IAG_BSR06-11, OGHA_04-064, HHSN271201300071C), and from various national funding sources is gratefully acknowledged. This study has been realised using the data collected by the Swiss Household Panel, which is based at the Swiss Centre of Expertise in the Social Sciences. The project is financed by the Swiss National Science Foundation. The United States Aging, Demographics, and Memory Study is a supplement to the Health and Retirement Study (HRS), which is sponsored by the National Institute of Aging (grant number NIA U01AG009740). It was conducted jointly by Duke University and the University of Michigan. The HRS is sponsored by the National Institute on Aging (grant number NIA U01AG009740) and is conducted by the University of Michigan. This paper uses data from Add Health, a program project designed by J Richard Udry, Peter S Bearman, and Kathleen Mullan Harris, and funded by a grant P01-HD31921 from the Eunice Kennedy Shriver National Institute of Child Health and Human Development, with cooperative funding from 17 other agencies. Special acknowledgment is due to Ronald R Rindfuss and Barbara Entwisle for assistance in the original design. Information on how to obtain the Add Health data files is available on the Add Health website. No direct support was received from grant P01-HD31921 for this analysis. The data reported here have been supplied by the United States Renal Data System. The interpretation and reporting of these data are the responsibility of the authors and in no way should be seen as an official policy or interpretation of the US Government. Collection of data for the Mozambique National Survey on the Causes of Death 2007-08 was made possible by USAID under the terms of cooperative agreement GPO-A-00-08-000_D3-00. This manuscript is based on data collected and shared by the International Vaccine Institute (IVI) from an original study IVI conducted. L G Abreu acknowledges support from Coordenacao de Aperfeicoamento de Pessoal de Nivel Superior (Brazil; finance code 001) and Conselho Nacional de Desenvolvimento Cientifico e Tecnologico (CNPq, a Brazilian funding agency). I N Ackerman was supported by a Victorian Health and Medical Research Fellowship awarded by the Victorian Government. O O Adetokunboh acknowledges the South African Department of Science and Innovation and the National Research Foundation. A Agrawal acknowledges the Wellcome Trust DBT India Alliance Senior Fellowship. S M Aljunid acknowledges the Department of Health Policy and Management, Faculty of Public Health, Kuwait University and International Centre for Casemix and Clinical Coding, Faculty of Medicine, National University of Malaysia for the approval and support to participate in this research project. M Ausloos, C Herteliu, and A Pana acknowledge partial support by a grant of the Romanian National Authority for Scientific Research and Innovation, CNDS-UEFISCDI, project number PN-III-P4-ID-PCCF-2016-0084. A Badawi is supported by the Public Health Agency of Canada. D A Bennett was supported by the NIHR Oxford Biomedical Research Centre. R Bourne acknowledges the Brien Holden Vision Institute, University of Heidelberg, Sightsavers, Fred Hollows Foundation, and Thea Foundation. G B Britton and I Moreno Velasquez were supported by the Sistema Nacional de Investigacion, SNI-SENACYT, Panama. R Buchbinder was supported by an Australian National Health and Medical Research Council (NHMRC) Senior Principal Research Fellowship. J J Carrero was supported by the Swedish Research Council (2019-01059). F Carvalho acknowledges UID/MULTI/04378/2019 and UID/QUI/50006/2019 support with funding from FCT/MCTES through national funds. A R Chang was supported by National Institutes of Health/National Institute of Diabetes and Digestive and Kidney Diseases grant K23 DK106515. V M Costa acknowledges the grant SFRH/BHD/110001/2015, received by Portuguese national funds through Fundacao para a Ciencia e Tecnologia, IP, under the Norma Transitaria DL57/2016/CP1334/CT0006. A Douiri acknowledges support and funding from the National Institute for Health Research Collaboration for Leadership in Applied Health Research and Care South London at King's College Hospital NHS Foundation Trust and the Royal College of Physicians, and support from the NIHR Biomedical Research Centre based at Guy's and St Thomas' NHS Foundation Trust and King's College London. B B Duncan acknowledges grants from the Foundation for the Support of Research of the State of Rio Grande do Sul (IATS and PrInt) and the Brazilian Ministry of Health. H E Erskine is the recipient of an Australian NHMRC Early Career Fellowship grant (APP1137969). A J Ferrari was supported by a NHMRC Early Career Fellowship grant (APP1121516). H E Erskine and A J Ferrari are employed by and A M Mantilla-Herrera and D F Santomauro affiliated with the Queensland Centre for Mental Health Research, which receives core funding from the Queensland Department of Health. M L Ferreira holds an NHMRC Research Fellowship. C Flohr was supported by the NIHR Biomedical Research Centre based at Guy's and St Thomas' NHS Foundation Trust. M Freitas acknowledges financial support from the EU (European Regional Development Fund [FEDER] funds through COMPETE POCI-01-0145-FEDER-029248) and National Funds (Fundacao para a Ciencia e Tecnologia) through project PTDC/NAN-MAT/29248/2017. A L S Guimaraes acknowledges support from CNPq. C Herteliu was partially supported by a grant co-funded by FEDER through Operational Competitiveness Program (project ID P_40_382). P Hoogar acknowledges Centre for Bio Cultural Studies, Directorate of Research, Manipal Academy of Higher Education and Centre for Holistic Development and Research, Kalaghatagi. F N Hugo acknowledges the Visiting Professorship, PRINT Program, CAPES Foundation, Brazil. B-F Hwang was supported by China Medical University (CMU107-Z-04), Taichung, Taiwan. S M S Islam was funded by a National Heart Foundation Senior Research Fellowship and supported by Deakin University. R Q Ivers was supported by a research fellowship from the National Health and Medical Research Council of Australia. M Jakovljevic acknowledges the Serbian part of this GBD-related contribution was co-funded through Grant OI175014 of the Ministry of Education Science and Technological Development of the Republic of Serbia. P Jeemon was supported by a Clinical and Public Health intermediate fellowship (grant number IA/CPHI/14/1/501497) from the Wellcome Trust-Department of Biotechnology, India Alliance (2015-20). O John is a recipient of UIPA scholarship from University of New South Wales, Sydney. S V Katikireddi acknowledges funding from a NRS Senior Clinical Fellowship (SCAF/15/02), the Medical Research Council (MC_UU_12017/13, MC_UU_12017/15), and the Scottish Government Chief Scientist Office (SPHSU13, SPHSU15). C Kieling is a CNPq researcher and a UK Academy of Medical Sciences Newton Advanced Fellow. Y J Kim was supported by Research Management Office, Xiamen University Malaysia (XMUMRF/2018-C2/ITCM/00010). K Krishan is supported by UGC Centre of Advanced Study awarded to the Department of Anthropology, Panjab University, Chandigarh, India. M Kumar was supported by K43 TW 010716 FIC/NIMH. B Lacey acknowledges support from the NIHR Oxford Biomedical Research Centre and the BHF Centre of Research Excellence, Oxford. J V Lazarus was supported by a Spanish Ministry of Science, Innovation and Universities Miguel Servet grant (Instituto de Salud Carlos III [ISCIII]/ESF, the EU [CP18/00074]). K J Looker thanks the NIHR Health Protection Research Unit in Evaluation of Interventions at the University of Bristol, in partnership with Public Health England, for research support. S Lorkowski was funded by the German Federal Ministry of Education and Research (nutriCARD, grant agreement number 01EA1808A). R A Lyons is supported by Health Data Research UK (HDR-9006), which is funded by the UK Medical Research Council, Engineering and Physical Sciences Research Council, Economic and Social Research Council, NIHR (England), Chief Scientist Office of the Scottish Government Health and Social Care Directorates, Health and Social Care Research and Development Division (Welsh Government), Public Health Agency (Northern Ireland), British Heart Foundation, and Wellcome Trust. J J McGrath is supported by the Danish National Research Foundation (Niels Bohr Professorship), and the Queensland Health Department (via West Moreton HHS). P T N Memiah acknowledges support from CODESRIA. U O Mueller gratefully acknowledges funding by the German National Cohort Study BMBF grant number 01ER1801D. S Nomura acknowledges the Ministry of Education, Culture, Sports, Science, and Technology of Japan (18K10082). A Ortiz was supported by ISCIII PI19/00815, DTS18/00032, ISCIII-RETIC REDinREN RD016/0009 Fondos FEDER, FRIAT, Comunidad de Madrid B2017/BMD-3686 CIFRA2-CM. These funding sources had no role in the writing of the manuscript or the decision to submit it for publication. S B Patten was supported by the Cuthbertson & Fischer Chair in Pediatric Mental Health at the University of Calgary. G C Patton was supported by an aNHMRC Senior Principal Research Fellowship. M R Phillips was supported in part by the National Natural Science Foundation of China (NSFC, number 81371502 and 81761128031). A Raggi, D Sattin, and S Schiavolin were supported by grants from the Italian Ministry of Health (Ricerca Corrente, Fondazione Istituto Neurologico C Besta, Linea 4-Outcome Research: dagli Indicatori alle Raccomandazioni Cliniche). P Rathi and B Unnikrishnan acknowledge Kasturba Medical College, Mangalore, Manipal Academy of Higher Education, Manipal. A L P Ribeiro was supported by Brazilian National Research Council, CNPq, and the Minas Gerais State Research Agency, FAPEMIG. D C Ribeiro was supported by The Sir Charles Hercus Health Research Fellowship (#18/111) Health Research Council of New Zealand. D Ribeiro acknowledges financial support from the EU (FEDER funds through the Operational Competitiveness Program; POCI-01-0145-FEDER-029253). P S Sachdev acknowledges funding from the NHMRC of Australia Program Grant. A M Samy was supported by a fellowship from the Egyptian Fulbright Mission Program. M M Santric-Milicevic acknowledges the Ministry of Education, Science and Technological Development of the Republic of Serbia (contract number 175087). R Sarmiento-Suarez received institutional support from Applied and Environmental Sciences University (Bogota, Colombia) and ISCIII (Madrid, Spain). A E Schutte received support from the South African National Research Foundation SARChI Initiative (GUN 86895) and Medical Research Council. S T S Skou is currently funded by a grant from Region Zealand (Exercise First) and a grant from the European Research Council under the EU's Horizon 2020 research and innovation program (grant agreement number 801790). J B Soriano is funded by Centro de Investigacion en Red de Enfermedades Respiratorias, ISCIII. R Tabares-Seisdedos was supported in part by the national grant PI17/00719 from ISCIII-FEDER. N Taveira was partially supported by the European & Developing Countries Clinical Trials Partnership, the EU (LIFE project, reference RIA2016MC-1615). S Tyrovolas was supported by the Foundation for Education and European Culture, the Sara Borrell postdoctoral programme (reference number CD15/00019 from ISCIII-FEDER). S B Zaman received a scholarship from the Australian Government research training programme in support of his academic career. ; "Peer Reviewed"

TABVLÆ RUDOLPHINÆ, QVIBVS ASTRONOMICÆ SCIENTIÆ, TEMPORUM LONGINQUITATE COLLAPSÆ RESTAURATIO CONTINETUR Tabvlæ Rudolphinæ, qvibvs astronomicæ scientiæ, temporum longinquitate collapsæ restauratio continetur ( - ) Einband ( - ) [Frontispiz]: Ulm um 1627. ( - ) Titelseite ( - ) [Widmung]: D. FERDINANDO II ROM. IMP. SEMP. AUG. GERM. HUNG. BOHEM. &c. REGI. . ( - ) DEDICATIO. ( - ) ASTRO-POECILO-PYRGIUM KEPLERIANUM, ASTRONOMIÆ ORTVM, PROGRESSVM. ( - ) INDEX CAPITVM ET PRÆCEptorum in has Tabulas. PRÆFATION IN TABVLAS RVDOLPHI foli. IN PARTEM PRIMAM TABB. ( - ) [Abb.]: Schema referendum ad CAP. XX. fol. 56. ( - ) [Karte]: NOUA ORBIS TERRARUM DELINEATIO SINGULARI RATIONE ACCOMODATA MERIDIANO TABB. RUDOLPHI ASTRONOMICARUM ( - ) IN TABULAS RUDOLPHI PRÆFATIO. (1) CAPUT I. (9) DE ARITHMETICA LOGISTICA, IN HIS TABULIS NECESSARIA. DE NUMERATIONE. (9) CAPUT II. DE ADDITIONE ET SVBTRACTIONE NVMERORVM TAM SIMPLICIUM, QUAM Logisticorum. (9) CAPUT III. (10) DE MVLTIPLICATIONE ET DIVISIONE LOGIstica visitata, pro hic Tabulis; et deHeptacosiade, cujus ope suffulti, seblevamur illis. (10) RATIO EXCERPENDI EX Heptacosiade. (4 [12]) CAPUT IV. DE LOGARITHMORUM ADDITIONIBUS ET SUBTRAktionibus Cossicis. (5 [13]) I. REGULA DE SPECIE Arithmetices. II. REGULA DE SIGNO exeuntis. (5 [13]) [2 Tabellen]: (1)Additionum Cossicarum formae. (2)Subtractionum Cossicarum formae. (5 [13]) CAPUT V: DE REGVLA TRIVM SEV PROPORTOINUM, OPE HEPTAcosiadis exercenda in numeris Logisticis, ad venandam partem proportionalem. (5 [13]) CASVS I. (14) EXEMPLUM PER SEXAGESIMARIAM SOLAM. EXEMPLUM PER QUAdrivicenariam solam. (14) EXEMPLUM PER COLUmellam utramque. (14) CASVS II. (15) EXEMPLUM EX SEXAGESIMARIA. EXEMPLUM EX QUADRIVICENARIA. EXEMPLUM PER DUAS COLUMELLAS COPU. (15) CASVS III. (16) EXCEPTIO. ALIUD CONSILIUM IN HAC EXCEPTIONE. (16) CAPUT VI. DE LOGISTICOVM NVMERORVM QVADRATIS, RADICIBVS ET MEDIO PROportionali inveniendis. (16) DE LOGISTICI NUMERI, UT QUADRATI, RADICE EXTRAHENDA, OPE HEPTAcosiadis. DE MEDIO PROPORTIONALI INTER DUOS LOGISTICOS INVENIENDO. (17) CAPUT VII. DE VSIBVS HEPTACOSIADIS ALIIS. (17) DE CONVERSIONE HORARUM ET MINUTORUM IN Tempora seu Partes & Scruplua Æquatoris, & vicissim. (18) CAPUT VIII. DE ORDINATIONE, CANONIS LOGARITHMORVM, MESOLOGARTHMORVM, ET ANTIlogarithmorum, in his Tabulis exhibit: Et quomodo sit excerpendus cujusque Arcus vel Anguli Logarithmus, quomodo Antilogarithmus: quomodo vicissim cujusque Logarithmi vel Antilogarithmi Arcus vel Angulus. (18) CAPUT IX. IN RECTANGVLO RECTILINEO, DATO ANGULO INTER LAtera, data et proportione laterum, determinare angulos reliquos. (21) COMPENDIA SEV CAVTIONES. (21) [Tabelle]: TYPUS OPERATIONIS. (22) CAPUT X DE TABVLA ANGVLI, EIUSQUE USU. (23) CAPUT XI. DE ALIO PECVLIARI USU CANONIS LOGARITHMORUM, præcique in STATIONum punctis indagandis. (23) [Abb.]: (24) [2 Tabellen]: (1)Pone ergo secundo, angulum C tantum, quantus prima correctioneprodiit, scii. 14°· 35'. (2)Eum igitur proba tertiâ iteratione processus. (24) DE ANTILOGARITHMORUM INTERPUNCTIONE & usu. (25) [Tabelle]: EXEMPLUM. Sint latera sublimis anguli, seu ardua (26) CAPUT XII. DE ASCENSIONIBVS RECTIS, MEDIATIONIBUS COELI, Declinationibus, & Angulis Eclipticæ cum Meridiano. (26) CAPUT XIII. DE AMPLITVDINE ORTIVA: ET DE DIFFERENTIA Ascensionali, ejusque Tabulæ Synopticæ usu. (28) DATO PUNCTO SPHÆRÆ quocunque Declinatione ab Æquatore, indagare ejus Amplitudinem Ortiva. (28) DATO PUNCTO SPHÆRÆ QUOCUNQUE, EIUSQUE DECLINAtione ab Æquatore; indagare ejus differentiam Ascensionalem sub data Poli altitudine. (29) [Tabelle]: Via posteriori (29) DATA POLI ALTITVDINE, PER DATI LOCI SOLIS DIFFERENtiam ascensionalem indagare tempus semidiurnum & seminocturnum, âdeoque dici artificialis longitudinem. (29) VICISSIM DATA LONGITVDINE DIEI ÆSTIVÆ LONGISSIMÆ, invenire altitudinem Poli. (30) CAPUT XIV. DE ANGVLO ORIENTIS, seu altitudine Nonangesimi, ejusque Tabula & usu in querendis Asc. obliquis, veletiam punctis Eclipticæ orientibus. (30) [Tabelle]: Ergo valet (30) DATO PVNCTO ECLIPTICÆ ORIENTE, PER EIUS CUM HORIzonte constitutum angulum indagare Asc. obliquam. (31) SED ET IPSVM PVNTVM ECLIPTICÆ ORIENS, PER ANGUlum ejus Horizonte datum vel sumptum, & per Asc. obliquam datam inquiri potest. (32) TANDEM DOCEBO, PER SOLOS LOGARITHMOS, SINE ULLIS Aliis Tab. computare & angulum arientis, & unâ ipsum punctum oriens, ex datâ Asc. obliquâ universaliter & exactè. DATA PROFVNDITATE LOci Solis sub Horizonte, inquirere distantiam ejus loci ecliptici à puncto oriente vel occidente, mediante angulo orientis. (32) [Tabelle]: Sit Asc: obliqua 346°.48'. Ergò est supra Horizontem, & ad occasum; quærendumque est latus Eclipticæ ab occasu usque in o. Ei verò respondet latus Æquatoris 13°.12', quantum sc: est ab 166°. 48' descensione obliquâ ad 180°, seu ad o. (32) DATA STELLÆ LONGITVDINE ET LATIDUDINE; SUB DATA Elevatione poli, invenire punctum Eclipticæ ei cooriens, mediante angulo orientis. (33) [Tabelle]: Esto Planeta ♂ in 2°.3°' ⩗ cum Latitudine 4°.40'. australi sub alt. Poli 56: quæritur punctum ei cooriens. Cùm Mars, oriente 2°. 3o' ⩗ sit adhuc infrà, ponam angulum aliquem eorum, qui 3 ⩗ sequuntur. (33) CAPUT XV. DE ÆQUANDO TEMPORE IN ÆQUALITATEM DIERUM naturalium & Tabulis huic rei in servientibus. (33) CAPUT XVI. DE REDVCTIONE TEMPORUM IN DIVERSIS LOCIS Numeratorum ad Meridianum harum Tabularum: & de Catalogo Locorum. (36) [Tabelle]: Augustâ Vindelicorum Madritum Hispaniæ censentur milliaria Germanica 200: Fides æstimationis sit penes viatores. Divisis 200 per 15, fiunt partes circuli magni 13°. 20'. Augustæ est A. P. 48°.22', Madriti 40°.45'. (40) [2 Tabellen]: (1)Strabo libro XV Geographiæ, Susis Persepolim numerat stadia 4200. Vt autem sciamus quot stadia faciant hoc loco gradum circuli magni, notandum quòd idem author à Promontorio Caramaniæ australissimo, quod fretum Sinus Persici constituit, ad Portas Caspias numeret 14400 Stadia. Alt. Poli illic est 25°.3°', hic 43°.3°'. Intersunt Gr. 18 sub eodem quasi meridiano, quia Strabo longitudinem Persidis ducit à Septentrione in Austrum. Si Gr. 18, patent 14400 stadiis, uni competent 800 stadia. Et si 800 stadia dant unum, 4200 dabunt 5°. 15'. Tot sunt Gradus Susis Persepolim. Est verò altitudo Æq. (2) ALIUD EXEMPLUM ET TYPUS operationis. (41) DE MAPPA MVNDI VNIVERSALI. (41) CAPUT XVII. DE REDVCTIONE ANNORUM MENSIUM ET DIERUM, qui apud alias Nationes in usu sunt vel fuerunt, ad Annos ante & post Christum, adque Dies Menses & Annos Iulianos, quibus hæ Tabulæ sunt accomodatæ. (42) DE TYPO ANNI CONFUSIONIS; ET UNA, ANNI ROMAnorum veteris Popiliani. (45) DE CONVERSIONE TEMPORUM ÆGYPTIACORUM Iin Iuliana. (46) DE CONVERSIONE TEMPORUM PERSICORUM, IN IULIANA harum Tabularum & vicissim. (46) DE CONVERSIONE TEMPORUM ARABICORUM ET TURCICOrum Hegiræ in Juliana, & vicissim. (47) [Tabelle]: Vt, reliquit Leunclauius in Pandecte suo historiæ Turcicæ notatum in fine diplomatis Sultani Amurathæ, Annum Transmigrationis (Hegiræ) Mahometis 991, diem 27 Silchidze, id est Dulhajati. Quæritur in quem diem cujus anni Iuliani is competat. (47) [Tabelle]: Vicissim, Anno Incarn: 1576, die 23 Decemb. seu X Cal: Ianuarias, in quem diem cujus anni Arabici ab Hegira competit? (48) DE APPLICATIONE DIERUM IN ANNIS; ÆGYPTIACO, & Armeniaco, fixis, ad dies Julianos. DE CHARACTERISMIS ET FERIIS ANNORUM & dierum. DE CYCLO SOLIS: (48) PER CYCLUM SOLIS PRODERE FERIAM DIEI IULIANI propositi. (48) FERIAM PRODERE DIEI IN ALIIS ANNORUM FORMIS & in Arabica. DE CYCLO LVNÆ SEV AVREO NUMERO (49) IN PARTEM SECUNDAM TABB. RUDOLPHI PRÆCEPTA: (50) CAPUT XVIII. DE TABVLIS EPOCHARVM ET MOTVVM MEDIORUM, ET QUOMODO COLLIGENDI SINT MOTUS MEDII ex his Tabulis, & loca singolorum Mobilium media assignanda. (50) [Tabelle]: Sint colligendi motus medij ad diem 24 Iulij anni 3993 ante Christum currentis, horam 0°.33'.26" post Meridiem Vraniburgicum æquabilem. Invenitur ergo Epocha proximè antiquior 4000. hinc ablato numero anni 3993 ante Incarn. currentis, qui per suprà dicta bissextilis est, relinquntur anni 7 completi. Quare operatio erit talis. (52) [Tabelle]: Natus est RVDOLPHVS Il. R. I. à quo Tabulæ istæ sunt denominatæ, Anno Incarn. 1552, die 18 Iulij, Hora 6°. 52' Viennæ Austriæ. Esto tempus æquabile. Epocha proximè antiquior, & minor, quippe post Christum, est 1500. Ergò (53) CAPUT XIX. DE CANONIBVS SEXAGENARIIS ET RATIONE COLligendi ex iis. (53) [Tabelle]: Desidero motum Solis in annis ante Christum 3992, mensibus ultimis à Julio de anno 3993, diebus ultimis Iulij 7. Horis 23.26. 34. Ergo anni 3600 dant l". 0'. 0', restant 393. Sic anni 360 dant 6'.0', restant 32 pIeni. (54) [Tabelle]: Exempli gratia, Mercurii Revolutiones integræ fiunt mensibus ternis; itaque in anno Iuliano communi fiunt quatuor, idest 24', & insuper Sig. 1. 23°.43'.15'', id est Sex. 24'.43' 15''. (55) QVOMODO FORMANDA SIT UNIUS CUISQUE EX SEPtem Planetis Anomalia media: (55) CAPUT XX. DE TABVLIS PROSTHAPHÆESEON, ET DE RATIONE EXCERPENDI EX IIS MOtus Anomaliæ, vel etiam æquationes Eccentrici. (55) COMPENIVM PER LOGARITHMOS SV MENDI partem proportionalem. (58) [3 Tabellen]: (1)In Genesi RUDOLPHI superius in venti sunt motus. (2)Sic in Saturno, erant (3)In Jove. (58) [4 Tabellen]: (1)In Marte. (2)In Venere. (3)In Mercurio. (4)In LUNA denique pro Anomaliæ solutæ motu coæquato, quatenus luna adhuc est similis planetis caeteris, essetque planè similis, si contingeret eam simul copulari Soli vel ejus opposito. (59) ADMONITIO DE LVNA. (59) INVENIRE ANOMALIAM ECCENTRI ALICVIVS PLAnetæ, vel per Anomaliam Mediam, vel per Anomaliam coæquatam cognitam. (59) DE EXCERPENDA ÆQUATIONE ECCENTRICA EIUSque partibus. DE EXCERPENDO LIMANDOQUE CVM INTERVALLO, tùm Logarithmo intervalli Planetæ. (60) COMPVTARE LOCVM, SOLIS QUIDEM VERUM, QUINque verò Planetarum, (ut & Lunæ pro Copulis) loca Eccentrica, in suâ cujusque Orbitâ. (61) CAPUT XXI. DE TABVLIS LATITVDINARIIS. (61) [Tabelle]: Vt in Genesi RVDOLPHI. Inventa sunt loca sic. (61) DE REDVCTIONE, CVRTATIONE, INCLINATIONE, Ejusque Mesologarithmo, excerpendis & limandis. (61) DE LOGARITHMO FORMANDO INTERVALLI CVRTATI; & de curtando ipso intervallo, si quis eo uti vult. LOCVM ORBITÆ AD ECLIPTICAM REDVCERE. (62) [Tabelle]: In Genesi RVDOLPHI erant Intervallorum Logarithmi (62) CAPUT XXII. DE PROSTHAPHÆRESIBVS ORBIS ANNUI; QUIBUS PLAnetæ locus tandem absolvitur. (63) PROPORTIONEM FORMARE INTERVALLORVM, SEV distantiarum, Terræ & Planetæ, à Sole. ANGVLVM COMMVTATIONIS ILLVM DEFINIre, in quo contingit Prosthaphæris Orbis, (Seu etiam in Inferioribus, Elongatio à Sole) per quamlibet datam proportionalem Intervallorum maxima. (63) EXCERPERE VEL COMPVTARE PROSTHAPHÆRESIN seu Parallaxin Orbis, per Angulum Commutationis & Proportionem Intervallorum. (63) [4 Tabellen]: (1)Vt in Genesi RVDOLPHI Imp. Quia in Saturno Angulus Commut. fuit 155°.49".13", Proportio Intervallorum 225784: cum 156° in margine Tabulæ & 220000 in fronte, invenio Prosth. orbis proximam 2°. 53', sed cum 230000 invenio 2.34. Erit igitur ea circiter 2.44. Sed sine Tabula sic ago. Proportio intervallorum 225784 dat ex Canone Logg. 6°. l'. ad summum. Ergò Prosth. Orbis quæsita, quia de Saturno agitur, est minor hoc arcu. Et quia additis 90, fit Commutatio 96°. 1' multò minor quàm 155°.49'; multò igitur minor erit Prosth. Orbis quàm 6°. l'. Sit 3°. o'. Ergò secundum Caput IX. (2)Sic in Jove. (3)Sic in Marte. (4)Sic in Venere. (64) [Tabelle]: Sic in Mercurio. (65) ELOGATIONEM PLANEæ à Sole definire, tàm cujusque temporariam, quam Inferiorum Maximam, cujusque Intervallorum proportionis. (65) INTERVALLVM INDAgare; Terræ & Planetarum quinque unius, ejusque, si detur, Logarithmum. (65) INDAGARE LATITVDInem Planetæ. (66) CAPUT XXIII. DIRECTORIVM GENERALE, EX PRÆMISSIS PRÆCEPTIS particularibus, expeditè computandi vera loca Planetarum quinque, secundùm & longitudinem in Eclipticâ, & Latitudinem ab eâ. (66) [4 Tabellen]: (1)In præceptis superioribus jam traduximus exemplum hoc per prima septem membra præcepti hujus. Igitur octavò, cùm fuerit Locus Eccentricus in Ecliptica, in (2)Nonò, cum Argumentis latitudinum excerpuntur ex sua cujusque Tabula latitudinariâ Inclinationum Mesologarithmi isti (3)Ab his summis sunt auferendi Logarithmi Commutationum, petendi ex Canone. (4)Hi ut Mesologarithmi, quæsiti in Parte Canonis Mesologarithmorum, produnt Latitudines veras. (67) [Tabelle]: Sint indaganda loca, Martis & Veneris ad annum 1590, diem 3/13 Octobris, hram quintam matutinam, quia MÆSTLINVS Tubingæ hoc momento vidit Venerem quasi sub Marte. Primùm computetur locus Solis, quia nobis ille opus est ad utriusque Planetæ locum. (68) ADMONITIO DE ABBREVIANDO HOC Calcula. (69) ALIA RATIO, SINE LOGARITHMUS, COMPVTANDI loca Planetarum quinque ex iisdem Tabulis: ut facilitas superioris præcepti pateat ex comparatione membrorum singulorum. (69) [Tabelle]: Iam angulus Commutationis est 53°.18'. 38". semissis ergò 26°'39" 19": cujus Tangentem 50199 multiplica in Quotientem. (69) [Tabelle]: Sic etiam in Venere, Anomalia Media 173°. 58'. o" dat distantiam in orbita 71915. In hanc multiplicata curtatio 77, absectis 5 à facto, efficit 55: quod ablatum ab intervallo, relinquit curtatum in Ecliptica Subordina interv. (69) [2 Tabellen]: (1)Iam pro Latitudinibus, divide sinus Commutationum (prolongatos mente 5 Cyphris) (2)Denique in Tangentes Complementa Inclinationum 1°.49'. (70) CAPUT XXIV. DE APSSIONIBVS, VTI VOCANT, QUINQUE PLAnetarum. Habitudines Inferioreum ad Solem, distinguere. (70) PROPORTIONEM INDAGARE, ARCVVM DIVRNOrum Eccentri, Solis & Planetæ. (70) CVILIBET ANOMALIÆ PLANETÆ SVOS COMMVtationis Angulos & Prosthaphæresin Orbis, seu in Inferioribus Elongationem assignare, in quibus is fiat Stationarius. (72) [Abb.]: In hac figurâ S. Solem repræsentat, O Terram, A Planetam unum ex Superioribus, vel econtario, A Terram. (72) [4 Tabellen]: (1)Sic diurnus Eccentrici Solis est, hac Anomalia Solis, 58' circ. Ergo summa E & B 89°. 31'. (2)Cùm autem 8 sit semissis de 17 priori correctione, patet, si in repetitionibus pergamus, nos per semisses correctionum ultimarum venturos ad 24°.29', 24°. 31'. Hic est angulus C correctus, quod licet probare. (3)Et cùm proportio Intervallorum ut Logarithmus ostendat angulum 27°. 25', ponatur C minor. (4)Cùm prima correctio demserit de positione 66. secunda 14½, erit ut 66 ad 14½, sic hoc ad 3 circiter, & fiet. (73) ALIA FACILIORI VIA COMMVTATIONIS ANGUlos illos addiscere, in quibus, stante unaqualibet proportione Intervallorum,fiunt Stationes; idque præterpropter. (73) QUO PACTO SINT DISCERNENDÆ STATIONES, PRIMA 6 Secunda: item, quomodo cognoscamus, recténe sumptus sit diurnus arcus & distantia Solis à Terra, in operatione pæcepti prioris? (73) QUOMODO COGNOSCATUR, NUM PLANETA SIT directus, stationarius an retrogradus? VTRVM MAIOR AN MINOR INCLINATIONE, FVtura sit Latitudo Planetæ. (74) VTRVM LATITVDO PLANETÆ CRESCAT, AN DEcrescat, annè consistat? (74) SEDIMIAMETROS PLANETARVM APPARENtes indagare. (75) DE PLANETARVM κρύψϵι OCCULTATIONE, ET Επιτολή Emersione ex radUS Solis, QUOS OCCASUS Ortúsque Heliacos et ab usu frequenti generis voce Poëticos appellant. (75) CAPVT XXV. DE LVNA SEORSIM, ET PRIMÒ DE ANOmaila SOLVTA: (76) DE ANOMALIA SOLUTA. (76) [8 Abb.]: (1)I (2)II (3)III (4)IV (5)V (6)VI (7)VII (8)VIII ([78]) [Tabelle]: IN ANOMALIA MEDIA (79) CAPVT XXVI. DE MENSTRVA LVNÆ ANOMALIA ET ÆQVAtionibus. (79) DESCRIPTIO TABVLÆ SCRUPP: MENSTRUORUM ET VAriationis etc: (83) QUOMODO PER VIAM INDIRECTAM, SECVNDVM INgenium Hypotheseos physicæ computandus sit locus Lunæ ad quodvis tempus propositum: (86) DESCRIPTIO TABVLÆ ÆQUATIONIS LVMInis seu compositæ. (87) QUOMODO PER VIAM DIRECTAM, ET ASTRONOmiæ veteri magis accommodatam, computandus sit Lunæ locus in Orbita ex his Tabulis? (88) CAPVT XXVII. DE LATITVDINE LVNÆ MENSTRVA, ejúsque Tabulis. (89) CAPVT XXVIII. DE PARALLAXIBVS LUNÆ. (92) [Tabelle]: Itaque punctum occidens quærendum est. Erit igitur (94) IN PARTEM TERTIAM TABB. RVDOLPHI PRÆCEPTA. (95) CAPVT XXIX. DE ECLIPSIBVS SOLIS ET LVNÆ EMINVS CONIECTANDIS. (95) [Tabelle]: Contingebat autem solstitium illa tempestate circa 28. Iunij. Ergo (96) DE CYCLO OBVIATIONVM ⨀ & Ω & ratione indagandi ex eo, diem in anno Iuliano, Conjunctionis medij loci Solis & Nodi Lunæ ascendentis. (97) CAPVT XXX. DE TABVLIS MOTVVM ⨀ ET◗ SVBSIDIARIIS. (97) [2 Tabellen]: (1)Haud multo diversus est usus subsidiariarum, in computando loco Lunæ ficto, quod ejusdem temporis exemplo docebo. (2)Hic si Summa dierum in Revolutionibus integris, quæ proximè minor est collecto tempore, deficiat plusquàm dimidio Revolutionis, utendum est proximè majori, & processus fit alius. (98) CAPVT XXXI. DE REQVISITIS AD COMPVTATIONEM ECLIPSIVM. (99) CAPVT XXXII. METHODVS ECLIPSES COMPVTANDI. (100) QUOMODO COGNOSCATVR EXACTVM Copulæ, seu Eclipticæ, seu cujuscunque, locúsque ejus in Solis & Lunæ Orbitis. (101) QVO COMPENDIO IN VICINA ALTERVTRIVS COpulæ, locus Lunæ fictus convertatur in verum. (101) DIRECTORIVM, QVOMODO EX PRÆMISSIS COMputandæ sint Eclipses Lunæ. (102) [Tabellen]: et in hoc tempus desinere debet diurnus, quo indigemus, quia Copula cadit ante meridiem loci Lunæ computati. (102) [2 Tabellen]: (1)Ergò cuym distet Luna ab Apogæo D. 3. H. 16: erit (2)Comprobato loco Lunæ in ipso momento Obscurationis maximæ sequuntur reliqua. (103) ECLIPSIS SOLIS, QVOMODO SIT COMPUTANDA UNIversaliter, in quantum scilicet pars quæcunque Hemisphærij Telluris ad Solem conversi interventu Lunæ; privatur lumine Solis vel toto vel in parte: Quodnam tunc sit tempus Obscurationis maximæ, quæ mora Vmbræ Lunæ in Disco Telluris, quæ duratio Eclipsiationis omnimodæ per universam Terram, quod initium finisvè utriusque, tanquàm Vraniburgi. (103) [2 Tabellen]: (1)vel etiam inter Logarithmos Semicirculi. (2)Nam quilibet ejus Gradus valet 15 Milliaria Germanica. (104) INQVISITIO ALTITVDINIS GRADVS ECLIPTICÆ NONAGESIMI AB ORIENTE. (105) DE LOCIS IN TERRA, QVIBVS OBVENIVNT PHASES PRÆCIPVÆ IN ECLIPSI SOLIS. (106) DE CALCVLO ECLIPSIS SOLIS AD CERTVM ALIQVEM LOCVM. (108) [2 Tabellen]: (1)Primùm itaque quæro distantias Centrorum Solis & Lunæ. (2)Secundò quæro distantiam duorum Lunæ situum. (108) CAPVT XXXIII. DE CONIVNCTIONIBVS ET OPPOSITIONIBVS ALIORVM Planetarum, & de ΕΞΕΛΙΓΜΟΙΣ & ΑΠΟΚΑΤΑΣΤΑΣΕΣΙ. (114) IN PARTEM QVARTAM TABVLARVM RVDOLPHI PRÆCEPTA. (116) CAPVT XXXIV. DE OBLIQVITATIS ECLIPTICÆ VARIATIONE. (116) SPORTVLA GENETHLIACIS MISSA DE TABVLARVM RVDOLPHI VSV IN COMPVTATIONIBVS ASTROLOGICIS: Cum Modo Dirigendi novo & NATURALI. (121) DE ERECTIONE THEMATIS Cœlestis. DATA SIDERIS LONGITUDINE ET LATITUDINE, ASCENSIONEM ejus Rectam & Declinationem computare. (121) [3 Tabellen]: (1)Quare aufer arcum à latitudine, restat 6. 39. 30 subtensus angulo. Sic ergò operor. (2)Locus datus rursum est Sept. latitudo verò Mer. & major, casus iterum IlI: ablatâ ergò illâ, restat 8. 47. Ergò. (3)Ablatâ ergò latitudine à 22. 24, restat subtensus 20. 35. (121) [Tabelle]: Consentit igitur hac in plagâ cum loco dato, Quare casus fit primus, & lat. l0. 22 addenda est ad exscriptum ex Declinationum columnâ 22. 24. 27, fiet subtensus 32. 46.27. (122) DATA ALTITUDINE SIDERIS CUJUS EST NOTA LONGITUDO ET Latitudo, indagare Distantiam ejus à Meridiano, & hujus, comparatione cum loco Solis, Horam. (122) [Tabelle]: Ecce Opus (122) QVOMODO VENIATUR IN COGNITIONEM ASCENSIONIS OBliquæ Horoscopi, & per eam Gradús orientis, caeterarumque ordine Domorum. (122) [2 Tabellen]: (1)Sic igitur operandum. (2).hujus Antilogarithmus ablatus ab Antilogarithmo altitudinis Poli, relinquit Antilogarithmum altitudinis ejusdem Poli super circulum Domus propositæ. (122) [2 Tabellen]: (1)In exemplo. (2).cujus Declinatio 22. l0. 30 Meridiana additur huic altitudini P. 29. 22, fietque. (123) DE DIRECTIONIBVS secundùm REGIOMONT ANVM: (123) [2 Tabellen]: (1)Distat ergò à Meridiano in ortum, arcu 41. 43. Opus itaque tale. (2)Hic sub Alt. P. 45 super Circulum Positionis Significatoris habet ad scriptum Angulum Orientis (123) DE DIRECTIONIBUS secundùm KEPPLERVM. (123) DATO NVMERO ANNORVM ÆTATIS, ASSIGNARE LOCA DIREctionis, quatuor Significatorum. (124) ELECTO LOCO, AD QVEM DIRIGENVS SIT SIGNIFICAT ORVM unus; seu dato loco Promissoris, veL ejus radij, invenire numerum Annorum, quibus is venit ad Significatorem. (124) DATO NVMERO ANNORVM alicujus Accidentis, electoque ejus & Promissore & Significatore, qui sit vel Horoscopus, vel Medium Cœli, vel Pars Fortunæ, corrigere tempus Nativitatis, & sic locum Significatoris. (125) NOTÆ ET ANIMADVERSIONES NON NULLÆ AD PRÆCEPTA TABUlarum RUDOLPHI. ( - ) TABVLARVM RUDOLPHI ASTRONOMICARUM PARS PRIMA, QUÆ COMMUNIS PLVRIBVS STELLIS VEL etiam aliis aliarum disciplinarum usibus. ([1]) [Inhaltsverzeichnis]: ([1]) [Tabelle]: HEPTACOSIAS LOGARITHMORUM LOGISTICORUM & Quadrantis Arcuum respondentium. (2) [Tabelle]: CANON LOGARITHMORUM ET ANTILOGARITHmorum, ad singula scrupula Semicirculi. (12) [Tabelle]: Tabula ANGULI, pro Prostaphæresibus orbis Annui. (20) [Tabelle]: Pars Canonis LOGARITHMORUM Gr. 10. pro latitudinibus quinq; Planetarum. (22) [Tabelle]: Particular Canonis ANTILOGARITHMORUM exactiorum, ad denarios secundorum, pro Eclipsibus. (23) [Tabelle]: Tabula Ascensionum Rectarum, Declinationum, & Angulorum Eclipticæ cum Meridiano. (24) [Tabelle]: Synopsis brevis differentiarum Ascensionalium. (25) [Tabelle]: Tabula Altitudinis Nonagesimi, seu Anguli Orientis, ad singulos gradus Altitudinis Poli, & ternos Eclipticæ, pro Parallaxibus. (26) [Tabelle]: Tabulæ ÆQUATIONIS TEMPORIS TRIPLICIS. (32) CATALOGUS LOCORUM EUROPÆ PRÆCIPUE, SED ET AFRICÆ ASIÆQUE NON NULLORUM, CUM DIFFERENTIA TEMPORARIA MERIDIANORUM AB URANOPYRGICO, ET POLI BOREI ALTITUDINIBUS: EX FIDE OBSERVATORUM & Observationum cœlestium, ubi haberi potuerunt; aut ex intervallis itinerariis, chartisque Geographicis recentissimis. (33) A, B (33) C (33) D - L (34) M (34) N - R (35) S (35) T - Z (36) SYNOPSIS ÆRARUM USU ALIUM, QUOTQUOT AD NOSTRAM NOTITIAM PERVENERUNT: SUNT AUTEM COMPARATÆ, SINGULÆ CUM SUIS ANNIS ANTE VEL POST INCARNATIONEM VERBI: assignata etiam usualia Annorum initia Mensibus & Diebus anni Juliani. (37) [4 Tabellen]: (1)TABVLA Reductionis Dierum anni Iluiani veteris, ad Dies anni gREGORIANI Novi, hodie usitati in plerisque partibus Orbis. (2)ROMANORUM JULIANORUM. (3)ÆGYPTIACORVM ET PERSICOVM. (4)ARABICORVM HEGIRÆ. (39) [5 Tabellen]: (1)TYPVS ANNI CONFVSIONIS qui finem imposuit anno Romano veteri: nec non Iulianorum primorum 49. vitioforum. (2)Tabula ostendens, quomodo Menses exotici Solares fixi hodie cohæreant cum Mensibus Anni Juliani. (3)TABELLA HEBDOMADICA, ad Feriam diei indagandam, Primum in anno JVLIANO, benefico CYCLI SOLIS. (4)Deinceps Calendæ Mensium usualum, sie responderum diebus mensis Juliani ex observatione hodierna. (5)Rursum per TRIACONTETERIDA in anno ARABICO vago Hegiræ. (40) TABVLARVM RUDOLPHI ASTRONOMICARUM PARS SECUNDA, PLANETAS SINGVLOS seorsim complexa, (41) [Abb.]: (41) [Inhaltsverzeichnis]: (41) SOLEM (42) [2 Tabellen]: (1)EPOCHÆ SEV RADICES. (2)MOTVS MEDII. (42) [Tabelle]: MOTVS MEDII in Annis expansis et collectis. (43) [Tabelle]: Tabula Æquationem SOLIS. (44) [Tabelle]: CANON Sexagenarius Motuum mediorum SOLIS. (47) SATURNUM (48) [2 Tabellen]: (1)EPOCHÆ SEV RADICES. (2)MOTVS MEDII. (48) [Tabelle]: MOTVS MEDII in Annis expansis et collectis. (49) [Tabelle]: Tabula Æquationem SATVRNI. (50) [2 Tabellen]: (1)TABVLA Latitudinaria SATVRNI. (2)Termini Stationum SATVRNI. (53) JOVEM (54) [2 Tabellen]: (1)EPOCHÆ SEV REDICES. (2)MOTVS MEDII. (54) [Tabelle]: MOTVS MEDII in Annis expansis et collectis. (55) [Tabelle]: Tabula Æquationum IOVIS. (56) [2 Tabellen]: (1)TABVLA Latitudinaria IOVIS. (2)Termini Stationum IOVIS. (59) MARTEM (60) [2 Tabellen]: (1)EPOCHÆ SEV RADICES. (2)MOTVS MEDII. (60) [Tabelle]: MOTVS MEDII in Annis expansis et collectis. (61) [Tabelle]: Tabula Æquationum MARTIS. (62) [2 Tabellen]: (1)TABVLA Latitudinaria MARTIS. (2)Tarmini Stationum MARTIS. (65) VENEREM (66) [2 Tabellen]: (1)EPOCHÆ SEV REDICES. (2)MOTVS MEDII. (66) [Tabelle]: MOTVS MEDII in Annis expansis et collectis. (67) [Tabelle]: Tabula Æquationum VENERIS. (68) [2 Tabellen]: (1)TABVLA Latitudinaria VENERIS. (2)Termine Stationum VENERIS. (71) MERCURIUM (72) [2 Tabellen]: (1)EPOCHÆ SEV RADICES. (2)MOTVS MEDII. (72) [Tabelle]: MOVS MEDII in Annis expansis et collectis. (3 [73]) [Tabelle]: Tabula Æquationum MERCVRII. (74) [2 Tabellen]: TABVLA Latitudinaria MERCVRII. (2)Termini Stationum MERCURII. (77) LUNÆ (78) [2 Tabellen]: (1)EPOCHÆ SEV RADICES. (2)MOTV MEDII. (78) [Tabelle]: MOTVS MEDII in Annis expansis et collectis. (79) [Tabelle]: Tabula Æquationum LVNÆ. (80) [Tabelle]. Tabula Scrupulorum menstrorum, eorumque; Logarithmorum, particulæ Exfortis, et VARIATIONIS. (82) [Tabelle]: Tabella VARIATIONIS demonstrativæ, quarta parte maioris quam Tychonica proxima; quam tamen Observationes Tychonis nonnullæ confirmare videntur. Deducitur autem ex appendice Gr. 132. 45. Elongationis ◗ à ⨀, ad Lunationes integras I 2, in anno fiderio. (83) [Tabelle]: TABVLA Æquationis LVMINIS, compositæ ex Æquationis Menstruæ proportione competente reducta, Particula exforte, et Variatione TYCHONICA. (84 - 85) [2 Tabellen]: (1)Tabula Latitudinis LVNÆ simplicis, una cum Reductione loci Orbitæ ◗ ad Eclipticam, quæ valent, Nodo Ω in Quadris existente. (2)Tabula exhibens portionem ipsam Latitudinis Menstruam. (86) [2 Tabellen]: (1)Tabula pro Augmentatione Latitudinis Menstrua. (2)Residuum Tabulæ exhibentis portionem Latitudinis Menstruam. (87) TABVLARVM RUDOLPHI STRONOMICARUM PARS TERTIA, DE ECLIPSIBVS SOLIS ET LUNÆ, ALIIS QVE PLANETARUM CONGRESSIBUS ET CONfigurationibus. (89) [Tabelle]: Typus Aurei Numeri, neque Politicus, neque Ecclesiasticus usualis, sedmere Astronomicus, serviens indagandis Mensibus Eclipticis in Methodo Anni Juliani. (89) [Tabelle]: CACLO OBVIATIONVM Solis Medii et Nodi Lunæ AScendentis in Periodo; minori Annorum 37. Julianorum cum dieblus 2, maiori verò Annorum 2828 Iulianorum exacta. Et ponitur in anno huius Periodi primo, ♂ ⊙ Ω fieri inipso articulo mediæ noctis, quæ inchoat primum Ianuarii, quamvis nulla Epocharum seu initiorum possibilium id exactè habeat. Dies autem intelliguntur currentes, et cum latitudine à media nocte antecedenti usque ad mediam noctem fequentem. (90) [2 Tabellen]: (1)TABVLA Subsidiaria Motuum Solis. (2)Canonion dierum in Mensibus Anni completis. (91) [Tabelle]: TABVLA Diurnorum SOLIS, cum Horariis et Semidiametris. (92) [Tabelle]: TABVLA Subsidiaria Motuum LUNÆ. (94) [Tabelle]: CANON Motuum Lunarium in Periodis Anomaliæ integris, per centum annos expansos, perque; Centenarios et Millenarios collectos. (95) TABVLA ficti Motus seu Elongationis Lunæ, à loco, in quo ipsa proximè Apogæ fuit vel erit, velut in mense vacuo: cum horario ficto, pro Syzygiis Luminarium indagantis, et pro computandis locis Lunæ ad tempus propositum, si addas Æquationes Menstruas. (96) [3 Tabellen]: (1)TABVLA Latitudinis Lunæ in Eclipsibus, cum Reductione loci Lunæ ad Eclipticam, vel Loci Solis eisque; oppositi ad orbitam Lunæ. (2)TABELLA Parallaxum et Semidiametri Lunæ, cum Horario eius vero in Copulis, à puncto fixo numerato. (3)TERMINI ECLIPSIVM. (98) [Tabelle]: LVNATIONVM seu Coniunctionum Solis et Lunæ EPOCHÆ. EPACTÆ in annis solutis. (99) [Tabelle]: CANON Sexagenarius Dierum. (100) TABVLARVM RVDOLPHI ASTRONOMICARVM PARS QVARTA, De Obliquitatis Eclipticæ, Præcessionis Æquinoctiorum et Latitudinis Fixarum Prostaphæresibus. (102) [Tabelle]: Epochæ Argumenti Obliquitatis et Prosthaphæreseos Æquinoctiorum forma quintuplici. (103) [2 Tabellen]: (1)TABVLA Motus Medii Argumenti Obliqitatis Eclipticæ, pro Forma Obliquationis quintuplici. (2)TABELLA CORRECTIONIS OBLIQVITATIS. (104) CATALOGUS STELLARUM FIXARUM MILLE, EX ACCURATIS TEYCHONIS BRAHE OBSERVATIONIBUS ET CALCULO AD ANNUM INCARNATIONIS MDC. COMPLETUM. (105) VRSA MINOR, CYNOSVRA. VRSA MAIOR, HELICE. (105) DRACO. (105) CEPHEVS. BOOTES, ARCTOPHYLAX. CORONA BOREA. ENGONASI, HERCVLES. LYRA, VVLTVR CADENS (106) OLOR, CYGNVS. (106) CASSIPEIA. PERSEVS. (107) AVRIGA, HENIOCHVS, ERICTHOIVS. (107) OPHIVCHVS, SERPENTARIVS. SERPENS OPHIVCHI. SAGITTA SIVE TELVM. AQVILA SEV VVLTVR VOLANS. ANTINOVS. DELPHINVS. QVVLEVS, EQVI SECTIO. (108) PEGASVS, EQVVS, ALATVS. ANDROMEDA. TRIANGVLVS, DELTOTON. COMA BERENICES. (109) PARS SECVNDA DE STELLIS FIXIS XII. SIGNORVM ZODIACI. (109) ARIES. (109) TAVRVS. GEMINI. CANCER. (110) LEO. (110) VIRGO. LIBRA. SCORPIVS. SAGITTARIVS. (111) CAPRICORNVS. (111) AQVARIVS. PISCES. (112) PARS TERTIA CATALOGI COMPLECTITUR FIXARUM, QUÆ XV. IMAGINES MEridionales offormant, à veteribus annotatarum partem potissimam. (113) CETE. ORION. ERIDANVS. (113) LEPVS. (113) CANIS MAIOR. CANIS MINOR, PROCYON. ARGO NAVIS. HYDRA. CRATER. CORVVS. CENTAVRVS, CHIRON. (114) IN HYDRA. IN CENTAVRO. IN LVPO. IN THVRIBVLO. IN CORONA AVSTRALI. IN PISCE NOTIO. (117) GRVS. PHŒNIX. INDVS. PAVO. APVS, AVIS INDICA. APIS, MVSCA. CHAMÆLEON. TRIANGVLVM. PISCIS VOLANS, PASSER. (118) DORADO, XIPHIAS. TOVCAN, ANSER AMERICANVS. HYDRVS. (119) [Tabelle]: TABULA REFRACTIONUM TRIPLEX, TYCHONIS BRAHE diutinis & multiplicibus Observationibus confirmata; potissimum in freto SUNDICO, quo mare Balthicum Oceano Germanico infunditur: pertim verò etiam in Regni Bohemiæ arce Cæsarea BENATICA: aëre defæcate, quàm fieri potuit, ad hoch electo. (119) Einband ( - ) Einband ( - ) Buchschnitt ( - ) Buchschnitt ( - ) Buchschnitt ( - ) Farbkeil ( - )